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What are conditional probabilities conditional upon?
Keith Hutchison
The British Journal for the Philosophy of Science; Dec 1999; 50, 4; Academic Research Library
pg. 665

Brit. J. Phi/. Sei. 50 ( 1999), 665-695 

What Are Conditional Probabilities 
Conditional Upon? 

Keith Hutchisan 

ABSTRACT 

This paper rejects a traditional epistemic interpretation of conditional probability. 
Suppose some chancc process produces outcomes X, Y . ... , with probabilities P(X). 
P( Y) . ... lf later observation revcals that outcome Y has in fact been achieved, then the 
probability of outcome X cannot normally be revised to P(XIY) [= P(X nY)/P(}')]. This 
can only be done in exceptional circumstances-when more than just knowledge of 
Y-ness has been attained. 

The primary reason for this failure isthat the weight of a piccc of evidence varies with 
the means by which it is provided, so knowledge of Y-ness does not havc unifom1 
impact on the probability of X. A better updating of the probability of Xis provided by 
P(XI Y* ), where Y* is not an outcome of the chance process being observed, but the 
sentence 'the outcome Y has been observed', an 'outcome' of the subsequent observa-
tion proces~. 

This alternative formula is widely endorsed in practice. but not weil recognized in 
theory. where the ovcrsight has generated some unsatisfactory consequences. 

1 The epistemic interpretation of conditional probahilitv 
2 Clearing away sonw undergrowth 

2.1 Non-epistemic Interpretations 
2.2 The dual chance-processes involved in epistemic conditioning 
2.3 ls the proposed 'new · rule really novel? 

3 The negative argument 
3.1 The basic chance-process 
3.2 Elaboration of the chance-process 
3.3 Refuting the Standard Fonnu/ation 

4 The positive supp!ement 
4.1 Correcting the Standard Formularion 
4.2 When does the Standard Fonnu/ation work? 
4.3 Updating in ignorance of epistemic procedure 
4.4 Updating upon receipt of irrelevant or incorrect information 
4.5 A revised model and formula 
4.6 Abbreviating the revised model 
4.7 An illustrative example 

5 Some traditional information-puzzles: Monty Hall 

t'; Oxford University Press 1999 



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666 Keith Hutchisan 

1 The epistemic interpretation of conditional probability 
The main thesis ofthis essay is very simple and negative. though supplemented 
by a more complicated positive sub-thesis. The core ofthe negative case argues 
(in Section 3) that there is something dramatically unsatisfactory with the 
standard epistemic interpretation of conditional probability (the 'Rule of 
Conditionalization ·, as it is often called), whilc the positive coda (Sections 
4-5) proposes a replacement, a subtle adaption of the Standard Rule. Though 
this modified rule is already widely used, the difference between the two 
fonnulations is not weil recognized. Before the negative case is developed, 

however, a number of preliminaries need to be dealt with (in Section 2) to 
clarify the problem-after this long introduction has sketched the road ahead 
in some detaiL 

Widely deemed to govem the updating ofprobabilities (in the various senses 
ofthat difficult word) on receipt of new information, the version of the Rule of 
Conditionalization that I object to is typically formulated as follows: 1 

Suppose circumstances are such as enable the reasonable observer of some 
chance-process to place 'confidence' P(X) in the outcome being of type X. 
THEN P(XI Y) ['=' P(XnY)!P(Y)] represents the confidence that observer should 

place in the outcome being of type X after the circumstances of the observer 
change. through addition of just the knowledge that some outcome of type Y 
has in fact been achieved. 

To sec what is meant by this Rule, it is useful to choose a concrete example. 
where there is Iittle confusion about the probability-values. even among those 
who espouse radically different notions of probability itself Suppose then a 

1 For examples of thc cpistemic Rule stated in forms that sccm equivalcnt to mine. see Skyrms 
[19~6]. p. IR9): Resnick ([1986]. p. 75); Kyburg ([1990]. pp. 50-2); Howson and Crbach 
( [ 1993]. p. I 04); Savage ([ 1954]. p. 44): Maher ([ 1993]. p. 85 ): Dudewic10 and Mishra 1[1988]. 
pp. 39-51 ). 

For a ven.ion of the rulc that differs importantly (yet erraticallyJ from the onc I treat as 
standard hcrc. see my discussion bclow (in Section 2.3) ofthe alternative formulation in terms of 
H ( 'hypothesis·) and F ( ·evidence · ). The subtle difference in meaning between the outcomc · }" 
here. and the vague 'E' there. is a key to this whole essay. 

A similar key is provided by the uscful treatment in De Finetti ([ 1974[. vol. I. pp. 134-51. 
More lcisurely than most. this discussion genuinely attempts to explain the rationale behind thc 
Standard Formulation. Two of De Finetti's rcmarks help to indicate whcre the view defended in 
this paper ditlcrs from the 'tandard one. Firstly. Dc Finetti says that 'every evaluation of 
probability" io. conditional upon 'the state of information in which [the ohscrvcr] tinds bim-
self -hut r argue below that it is conditional upon hi' epistemic histor> (even if he does not 
know that history). Secondly. De Finetti motivates the Standard Formulation by envisaging a 
bctting game in which every debt is called off 'ifthe [conditionj does not turn out to he true'. This 
stipulation doe-. not yield epistemic conditionals, however. For De Finctti does not teil us what 
happcns when the condition is true but not known to bc true. To get the Standard Formulation, 
one needs to add something omitted hy Dc Finetti. hut difficult to comply with: that the debt is to 
be enforccd every time the condition turns oulto he truc (whether known tobe true or not). I An 
analogous requirement is introduced by in Section 4.2. when I observe that thc Standard 
Fonnulation prcsumö a 'persistent' observation-process.) 

For acceptable epistemic conditionals. however, one needs a more acccssible rule: the dcbt has 
to be enforced if ancl only if the condition is known to he mel. 



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What Are Conditiona/ Prohabilities Conditional Upon? 667 

family is chosen from the population of all two-child families in some region, 

and chosen 'randomly' -in thesensethat each family has equal probability of 

being chosen. (For simplicity, I shall use the word 'random' in this standard 
sense all through the discussion below.) The prohability that this family has 

two boys in it is approximately ± (while the probability that it has at least one 
boy is i). Suppose that after the choice has been madc. it is somehow ascertained 
that the family has at least one boy in it (without anything eise of relevance 

being discovered). Should the probability that the family has two boys now he 
updated? If so, has enough information been provided to allow a rcliable 

revision? And if that is so too, how is the updating to be effected? 

This example has been chosen to be one of those which precisely fits the 

Rute of Conditionalization as setout above; and that Rule applies immediately. 
It answers the first two of our questions in the affirmative: and provieles an 

algorithm that determines the updated probability for us. It gives us indeed no 

choice but to accept { as the updated prohability. 2 Forthis * is simply the value 
here of the formula's P(XnY)IP(Y), with X the outcome ('two hoys') of the 

initial draw whose probability is tobe revised. and Y the outcome asccrtained 
('at least one boy'). 3 

Arethese three answers good ones, however? I presume we all agree that the 

first of them is correct. 4 But the other two answers are \'erv suspect.Indeed. the 
dual function ofthis cssay is (firstly) to demonstrate that both are wrong. in that 

additional information is required to perform the updating reliably; then 

(secondly) to proviele a different, and weaker, rule, that uscs the appropriate 
extra information, to proviele a convincing updating. Both these points have 

been suggested clsewhere (in. say, the works cited in fns Rand 15)-hut with 

varying degrees of conviction. My task is to purse the question systematically. 

To catch a preliminary glimpse of why I take these views. suppose that the 

existence of a boy is revealed by simply randomly chom,ing a child from the 

selectcd family. Then in a long run ofsuch trials, halfthe families where a boy is 

revealcd will contain two boys. So it seems that a rational observer. who finds 
out both that and hmt· a boy has been detected, should now allocate the value ~ to 

the probability that the family has two boys. On the other hand, suppose the new 
knowledge had been acquired quite differently. by someone revealing the 

existence of a girl whenever possible. Such an informant would only indicate 

that the family contained a boy if it were impossible to indicate that it contained 

2 For some endonement.1 of this ans wer, sec Hodges and Lehmalm ( [ 1964[. pp. 7R-9). plus fn. 7 
below. Thc source' provided in fn. 7. howcver. need tobe rcad in conjunction with those cited in 
fn. 9. For some rejections of this ans wer. closely related to those pursued in thc prcsent essay 
ltogether with pointers to further endorscments of the 1 answer), see fns 6 and ~. 

' For here: PtXnY) = 1 (thc probability-prior-to-the-new-lnformation of the family BOTH contain-
ing two boys AND at least one hoy). while P(}} = 1 (the prnbahility-prior-to-the-new-infonnation 
of the state of affairs actuallv rcvealed bv that information). 

" lndccd. my remarks bclow simply do not address those-propensity-theorists. perhaps''-who 
believe the new information has no effect on the probability of two boys. 



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668 Keith Hutchisrm 

a girl. So after thus finding out that the chosen family contained one boy. the 
probability that it contained two of them would surely j ump to 1. 

Whether or not these really are the best answers to our question here has 
been the subject of much debate (see e.g. the works cited in fns 5, 6, 7 and 8). 
and it is perhaps not essential to agree with them yet. What immediately 
matters is: to feel the pull in their direction; to note that they provide a good 
evaluation of the long-run frequencies; and to accept the clash between them 

and the postcrior probability (*) calculated using the standard Rule of Con-
ditionalization. Yet that unreliable probability was the one imposed by the 
traditional algorithm, and we can now see why it is suspect: its observer did not 
know how the extra sex-information was being supplied. Finding out how the 
information was provided (apparently) enabled the observer to replace a 
defective estimate of the updated probability with a better one. 5 

Such confticting probabilities are very common. Indeed it will turn out that it 
is only in quite special circumstances that the Standard Formularion of the Rule 
provides acceptable posterior probabilities. For conditional probabilities of the 
form P(XnY)IP(Y) in fact serve primarily to tell us somcthing else, vi:::.: 'How 
are we to update the probability of two boys if we find out that we were wrong 
about how the family was being chosen, in that it was not chosen at random 
from the whole population of [two-child] families, but only from a sub-

population, that of such families with at least one boy?' (The * above is 
obviously the correct answer to this question.) 

The basicproblern with the Rulc ofConditionalization isthat the probability 
assigned by that Rule (here, the ~) is rarely the same as that tobe allocated after 
ascertaining the family belongs to the sub-population. if in truth the random 
choice was made from the whole population. For although arrival of the 
information does indeed indicate that the family was chosen from the sub-
population, it does not provide the slightest reason to believe that it was chosen 
randomly from that sub-population. (And usually it is not, as I demonstrate at 
length below, for information is normally-if not always?-biased, and this 
bias in effect de-randomizes the earlier choicc.) So the Rule needs to be 
reformulated if it is to be successfully applied to the epistemic circumstances 
set out in its formulation-situations where partial information is received 
about the outcome of a prior chance-process. 

1t is certainly not enough to insist that the Standard Rule is only meant to 
apply to situations where our information arrives free of bias (as hinted by at 
least one commentator-after glimpsing the inadequacies of the Standard 
Rule). 6 Firstly because bias is context-dependent, so information seems 

5 This is the basic message of Nickerson (j 1996 j). For other similar messages. see also fns 8 and 15. 
"Cf Falk ([1993]. pp. Sn, 177-8). who gives the impression that thc probahility oftwo hoys in a 

two-child family after discovering at least onc boy would be * if the source was not tainted by 
sexist attitudes. 



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What Are Conditional Probabilities Conditional Upon? 669 

never to be bias-fee simpliciter. Secondly, because it tums out that the 
Rule only works when there is bias (but the bias has to be appropriate). 
In consequence, to impose this restriction would be to give up too hastily. 
For successful conditioning can be routinely effected with biased 

information. 
Indeed, what we need to do instead is just reformulate the Rule gi ven above 

so that it accords with prevailing practice. For despite the lip-service paid to the 
Rule in theoretical discussion, epistemic conditionals are often evaluated 
differently in concrete situations. They are then evaluated using a slightly 
different rule, inadequately distinguished from the articulation above. This 
alternative rule happily accommodates biased information; and it uses more 
data than the Standard Rule, since it typically rcquires that we know both 
P(XnY*) and P(Y*), where the Y* is not an outcome of the chance-process 
which takes place prior to the observation. Jt is instead an outcome of a 
subsequent observation-process, vi:::., the epistemic proposition 'the observer 
has ascertained the original outcome is of type Y'. The revised rule then assigns 
an updated probability given by P(XnY*)/P(Y*). 

To see this alternative rule in action, Iet us return to our family example, 
supposing that after selecting the family a girl is revealed to the observcr 

whenever possible (i.e. whenever the family contains at least one girl). Other-
wise (i.e. when the family contains two boys) a single boy is revealed. The 
information hcrc is clearly biased against revealing boyness. but that does 
not hinder us from updating the probability of there being two boys ( or girls) in 
the family, after detecting one boy (or one girl). Indeed, an observer who 
understands how the information is being released, should allocate a prob-
ability of ~ to there being two girls in the family if he finds out there is at least 

one girl. 
This ~ (as we have seen) is the answer given by the Standard Formulation, _, 

despite the bias. (lt provides, indeed, a preliminary illustration of what I 
meant above by ·appropriate' bias.) But it is also the answer given by the 
Y*-alternative, though I will not pause to perform the calculation, for the 

Y*-alternative is more interesting in the case of boy-information. For our 
observer should obviously reject this standard answer ~ (for the prob-
ability of two boys) aftcr finding out by this means there is a boy in the 

family. 
Indeed, it is clearly a certainty that there are two boys, ifthe observer is told 

there is one. This, however, is the posterior probability provided by the 
alternative formula. For the Y*-formula recommends an updating to !1~, 
since both P(Y* )-the probability of finding out that there is at least one 
boy-and P(XnY*)-the probability of BOTH there being two boys AND 

finding out there is a boy in the family-are !· 
Similarly, in our earlier example (where the sex-information was released by 



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670 Keith Hutchisan 

making a random selection), P( Y*) = ~ and P(XnY*) = ~· so the alternative 
formula updates the probability to ~· the figure we earlier recommended (as 
according with the long-run frequency). 

The fact that we have reached updated probabilitics hcrc that diffcr from 
those provided by the Standard Formularion of the Rulc of Conditionalization 
(and will similarly meet many more below, as summarized in Table I in 
Section 3.3) does not directly refute that Rule, for morc information was used 

in reaching the non-standard value than the formulation under attack allows. 
The observer did not just ascertain that there was at least one boy in the family, 
for he was also presumed to have understood how that informationwas given 
to him. 

But we have certainly indicated that the updated probability varies with the 
process by which the information is received, for we have seen that the 

comparable information had different effects on the updating in the case of 
girls and boys. So the 'impact' of a piece of information varies with the means 
by which that information gets acquired. The Standard Rule ofConditionaliza-
tion gives no hint that this is so, and does worse: it seems to preclude its 
accommodation. For the Rule's insistence that nn additional information reach 
the observer beyond knowledge that an outcome Y has been achieved means 
that the pre-conditions for application of the Rule are breached whenever 
reccipt of the new information brings with it the slightest hint as to how it 
arrived! 

We have, however, glimpsed an alternative formula, one which seems 
capable of accommodating such surplus information. Indeed, the alternative 
demands it, for its use requires one to know P( Y*) and P(XnY* ), and these 

depend on the epistemic procedures. In consequence, the revised rule gives 
better evaluations of the updated probability. 

A thoroughly satisfactory description of the process of conditioning is, 
however, difficult to formulate. One obstacle to its articulation will tease us 
all through the discussion below. the apparent need to insist that the Y (or Y*) 
here represent the totality of relevant new knowledge-as implied by my 
including the word 'just' in our initial formulation of the Standard Rule of 
Conditionalization. Another reflects the fact that it is not, in the end. essential 
to conditioning that the observcr acquirc knowledge of some outcome of the 
chance-process: onc can rationally update probabilities after receiving claims 
about outcomes that are known tobe inaccuratc: and onc can do the samc on 
receipt of knowledge about matters other than the outcomes of the chance-

process under investigation. Such qucstions will bc pursucd further bclow, 
where they eventually Iead us to entertain a revised formula for the posterior 
probability slightly different to the Y*-formula above. (We abandon the 

latter formula only because it is not of wide enough scope. not because it is 
wrong.) 



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What Are Conditional Probabilities Conditional Upon? 671 

Another quite different set of difticulties arises because the interpretation 
of the words and symbols used to express the conditioning process are much 
at issue-varying with views as to the nature of probability. Some might 
interpret the 'contidence' as nothing more than an estimate of the long-run 
frequency: others will sec it as a measure of the weight of evidence 
available to the observer. Some will interpret the X and Y as sentences: 
others will interpret them as sets, outcomes perhaps of a chance-process. 
Fortunately, the essence of the case here can be presented without choosing 
between these alternative interpretations, for the argument develops out of 
those simple urn models at the centre of our understandings of probability, 
models where (it seems) the variety of interpretations is of least signiticance. 

The case does not involve exotic examples from the fringes of our under-
standing-those inconvenient examples where the disputing interpretations 

drift apart. 
Indeed, my focus here will be on core examples of the conditioning 

process, the cases where there is a minimum of conceptual confusion, 
where prior probabilities are clearly detined, and where frequency-values 
(and safe betting ratios) are widely deemed an acceptable guide to prob-

abilities. For an attack on the Standard Formulation is all the stronger. 
the more it relies on unproblematic chance-processes. Accordingly, it is 
unlikely that the probability values I cite below will be resisted, for (as 
already hinted) my primary target here is not false estimates of probability. 
What 1 seek is a better analysis of the procedures we use to reach those 

probabilities. 
As a consequence of this emphasis on articulation. I am not directly 

interested here in recommending novel probability assignments. except in 
the circumstances where too literal a reading of the Rule has triumphed over 
common sense. A good example of such a retreat from common sense is 
provided by the widespread claim that the probability of our randomly 

chosen two-child family having two boys must always change from ± to * 
upon discovery that the family contains a boy. As already observed, this 
grossly incorrect result comes from direct application of the Standard 
Rule of Conditionalization. and (yes 1) is sometimes cited as an example 
of the way thcory functions to correct the erroneous common sense of the 
beginner 17 

7 E.g. Isaac ([1995]. pp. 2-'1-5). Cf. Feiler I[ 1968[, vol.l, pp. 11-'1-8. 125). Note especially the 
warning (at Feller"s p. 116) that these interpretations are not to be taken too seriously. and 
compare Billingsky ([ 1986], p. -'1-'18). 1t is not initially clear that these discussions are intended to 
include epistemic conditionals. To confirn1 that this is a reasonable reading. one needs also to 
consult the discussion cited in fn. 9 below. 



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672 Keith Hutchisan 

More fully, my conclusions can be surveyed in the following (overlapping) 
terms.x When P(Y) -=/= 0, 

a) If P(XIY) is made equal to P(XnY)/P(Y) by fiat (via definition, axiom, etc.). 
then P(XIY) cannot be given the epistemic interpretation need for the Rule of 
Condi tional izati on. 

b) For P(XnY)/P(Y) is usually not a good estimate of the confidence to be 
placed in X after discovery that Y is true. 

c) Yet P(XnY)/P(Y) is sometimes a good estimate of the confidence to be 
placed in X after discovery that Y is true-but only in circumstances where 
more has been ascertained than just thc truth of Y. So for P(XnY)IP(Y) to 
yield the probability posterior to the discovery it is oftcn important that Y not 
be the 'total evidence ·. 

d) If indeed one knows that the process which provides the knowledge of Y is 
'persistent' (in a technical sense elaborated below). then P(XnY)/P(Y) is a 

good estimate of the confidence to be placed in X after discovery that Y is 
true. 

e) More generally. the confidence tobe place in X after ascertaining Y is not a 
function of the variables X and Y, but is a function of X and a somewhat 
different variable, one representing the circumstances under which the truth 
of Y was conveyed to us- Y* above. 

The essence of my case lies in the final claim here. e). for once that is made 
clear, the other sub-theses will be relatively obvious. In order to establish e). 
however, we need to confront a conspicuous haziness in the way the traditional 

Rule of Conditionalization was expressed above. where there was (as already 
noted) no extended accommodation of the 'principle of total evidence'. the 
idea that Y needs to represent 'all' the evidence that becomes available to us. 
Though this principle is partially denied in sub-thesis c). a corrected version of 
it remains essential to all updating of probabilities. yet we also need a version 
which allows s(nne of the new evidence to be ignored-that which is irrele-
vant. (Otherwise we are quickly led into an infinite regress.) But there are 
severe difficulties in articulating a criterion for relevance. so the principle 

s The argument ofthe present paper overlaps somewhat with those in: Bar-Hillel and Falk ( [1982]): 
Freund (I 19651J: Korb ([1994]): :-.lickcrson ([ 19961): Shafer ([ 1985 j): I plus some further Iiterature 
cited in these sources). But there are rnajor differences. S<) Shafer seeb to defend the Standard 
Formula for PI XI Y). and thus finds a protocol that saves it -thc onc l refcr to below as ·persistent' 
Iransmission: Shafer does not seek to emphasiLe that the formula is false for most protocols: Shafer 
focuses overwhelmingly on subjectivc probabilities. and barely appreciates (e.g. p. 269) that the 
same problern occurs for frequentisK :-.lickerson and Bar-Hillel and Falk by contrast ignore thc 
Standard Formula: in consequence. they do not dcvclnp thc contrast between "ontic" conditionals 
(to which that formula applies) and epistemic ones: neithcr do they develop a substitute forthat 
formula. But they fully appreciate that posterior probabilities vary with thc mcans by which a piece 
of information is received. Korb and Freund certainly recognize thü, too. hut Korb"s primary 
interest lies elsewhere. whilc Freund presents his argument as an attack on something like the 
principle of indiiTerence ( 'the assumption of equal a posteriori probabilitics·. p. 29). See also fn. 15. 



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What Are Conditional Prohabilities Conditional Upon? 673 

remains imperfectly formulated, here and elsewhere. Accordingly, we will 
continue to proceed via concrete example, for once this example is understood, 

generalization will be reasonably obvious, and a negative case of wide validity 
can be made without so\ving the problern ofidentifying total relevant evidence 
in general. As already hinted, we eventually see that the relevance of the 
information is not what matters, in that it is often rational to revise probabilities 
after the receipt of information which, by normal criteria, is irrelevant. 

But before we move to the core of the case, several minor pieces of house-
keeping seem to warrant attention. 

2 Clearing away some undergrowth 

2.1 Non-epistemic interpretations of conditional probability 
Conditional probabilities are often interpreted hypothetically, and even coun-
terfactually, P(XI Y) being (say) the probability to allocate X were it to happen 
to be the case that Y was true. It is important to observe that such ·ontic · 
interpretations (as wc shall call them below) are not being pursued in the 
present essay-whose concern is with investigatory situations. where the truth 

of Ybecomes ascertained or believcd, etc. For there are many circumstances in 
which the two types of conditional probability are dramatically different: 
Bovine Spongiform Encephalitis can (at the moment of writing) only be 
diagnosed via a post-mortem, so the probability that a cow known to be 
infected with BSE will be alive tomorrow is zero. But it is widely believed 
that many living cows have the disease. and can survive for long pcriods of 
time with it. The ontic conditional is a dubious guide to the epistemic one here. 

Nor are we concerned here with another class of questions, also seen as 
exemplifying conditional probability, those which estimate the probability of 
some fact or observation, if a particular scientific theory that bears on the fact is 
true (and vice versa). So I do not ask how we estimate the probability that a cow 
with BSE will survive one year after infection, if a particular speculation about 

the operation of BSE happens to be true. 
Though such conditionals are not the target of thc criticisms below. the 

discussion does have an indirect bearing on them, in part because of an ongoing 
tendency in the Iiterature to blur the distinctions betwcen the varieties of 
conditionals. 9 And more importantly, though the problems we are directly 
concerned with seem to have a different character, we will eventually conclude 
that epistemic conditionals are simply special cases of ontic ones-with the 

0 See e.g. Kyburg ([ 1990] p. 50. where conditional probabi1ity is describcd epistemically, but 
stochastic indcpcndencc interpreted ontically): Resnick (] 1986], pp. 48, 75. whcre conditiona1 
probability is introduced ontically, then-1ater-interpreted epistemically. and pp. 53-4. where 
the author oscillates between ontic and cpistemic intcrprctations): Dudewicz and Mishra ([ 1988]. 
pp. 38-50. where questions about conditiona1 probabi1ity are indiscriminately ('') pw.ed in 
epistemic and ontic terms ). 



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674 Keith Hutchisrm 

chance-process specially modified to include the epistemic activity that pro-
vides the partial knowledge used for updating. Indeed. it should already be 

obvious that the epistemic conditional which is the probabilitytobe assigned to 
X when Y has been ascertained is just the ontic conditional which is the 
probabilitytobe assigned to X when Y* is true. (This is tobe contrasted with 
P(XnY)IP(Y), the probabilitytobe assigned to X when it is only true that Y is 
true.) 

2.2 The dual chance-processes involved in epistemic 
conditioning 

It is important to note (with e.g. Rosenkrantz [1977]. pp. 48-52) that as soon as 
we pose the sort of question the Rule of Conditionalization is routinely 
supposed to answer. we are envisaging two quite separate types of activity 
as taking place. The first is typically a chance-process in which a physical 
system evolves 'non-deterministically' from some initial macrostate to one of 
a range of final states-like the throw of a die which ends up in one of six 
possible orientations on the tablc below. (The letters X and Y without the 
asterisks have been used above to refer to final states. 'outcomes', of this initial 

process.) 
The second proccss is typically subsequent to the first, and is an information-

process, conveying a message to some observer that bears on the outcome 
achieved in the first process. lf (say) our die had dots on it of more than one 
colour, an example of this second type of process would be the discovery that 
the dots on the uppermost face of the die were all red. As a result of the 
supplementary information provided by this second process. the observer 
might weil abandon the probabilities he initially allocated to the outcomes 
of the first process and replace them by updated probabilities. In our case. 
suppose (tobe concrete) that the die had red dots for 1, 2 and 3; and bluc dots 
for 4, 5 and 6. An observer who discovered that red dots had been thrown might 

then revise thc probability of I or 2 having been thrown from { to ~­
Probabilities prior to the second process are supplanted by those posterior to 

it. That is what conditioning is about. (More precisely. that is what condition-
ing seems tobe about: we eventually see that it is not important that the second 
process be revealing facts about the outcome of the first one.) 

Of course. the second process here, the information-process, may itself 
involve unpredictable activity, and thus constitute aseparate chance-process; 
and even if the observing activity is not probabilistic, it can still be so 
conceptualized-since deterministic processes are simply chance ones with 
probabilities 1 and 0. The things I have designated Y* above are the outcomes 
(in the technical Kolmogorov sense) of this second chance-process. So the 
whole of our discussion here is set paradigmatically in a context where two 



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What Are Conditional Prohahilities Conditional Upon? 675 

distinguishable chance-processes are taking place: an 'evolution-process' (as I 
shall often call it below), and then an 'observation-process'. 

I am not sure that these separate processes are always easy to distinguish, 
or that the observation has to take place after the evolution. But these 
conditions are often met, and characterize the conceptually simpler cases 
of conditioning. Such cases can serve as type-specimens for more complex 
situations, and accordingly we concentrate on them here. For the fact that the 
Standard Formularion of the Rule of Conditionalization fails to provide the 
simplest of probability updatings suffices to rob that formulation of its 
credibility. To make the negative case. there is no need to Iook at blurred 
situations. 

It is important to separate the two different processes set out above. 1 0 

because the main problern I detect with the Standard Formtdarion is its implicit 
claim that the probabilities associated with the second of the pair, the observa-
tion-process, are irrelevant to the updating of probability. 11 Its formulations 
routinely assert that posterior probabilities can be calculated from nothing 
more than a knowledge ofthe probabilities associated with the first process. the 
temporal evolution. the one that undergoes observation. For they declare that if 

the second process detects that the first process has produced an outcome of 
type Y. then the new probability for an outcome of type Xis simply P(XnY)/ 
P( Y), where P(XnY) and P( Y) are both probabilities of outcomes of the first 

process. 
In contrast, I claim we need to attend to the probabilities associated with the 

outcomes Y* of the second chance-process, the P( Y*) as I have called them, the 
probabilities of receiving the conditioning information. The need to embrace 
these probabilities is widely recognized in practice, for computations of 
posterior probabilities do in fact often use understandings of the second 
chance-process (as we will see in the next section). But the fact that it is 
these probabilities which are being used is often obscured, for the distinction 
between the two processes is blurred, especially in theoretical discussion. 

So the main problern confronted in this discussion is one of articulation, 
finding the right description of procedures that are already in partial use. 

1
" Two of thc cxamples I give in Section 2.3 below of mis-application of the Rule of Conditio-

nalization both im olve situations whcrc the first chance-pmcess. that generating the priors. is 
not spccificd in any detail, and more or less left to the imagination of the rcadcr. This indeed is 
one reawn that prior probabilities are so oftcn clusive. and (in consequence) so widely 
distrusted. Yct when the first process is clearly spelled out, thc priors are quite unproblematical, 
and should raisc no hackles at alll 

11 For some discussions of conditioning. where it is reasonably clear that the probabilities of the 
obsen ation-proccss are overlooked. see: Rosenkrantz ([ 19771. pp. 48-9, cxample I). wherc thc 
protocol governing the stooge·s releao,e ofinformation is not mentioncd: Resnick ([1986]. p. 56, 
problem l ). where a 'persistent' proccss-in the sense set out at in Section 4.2 below-is 
prcsumably intended. but no indication is given that this is vital to the discussion; Dudcwicz & 
~!ishra (j 1988J, pp. 38-50). wherc no hint is given that thc mode of observation is relevant. 
Contras! the discussions cited in fns 8 anJ 15. 



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676 Keith Hutchisan 

Distinguishing the two chance-process involved m paradigmatic cases of 
conditioning is a vital first step in this analysis. 

2.3 Is the proposed 'new' rule really novel? 
There is, however, a popular formulation of the conditioning process that 
might seem to have solved the problern of articulation here, the formulation 
phrased in terms of a 'hypothesis' (H = 'the family has two boys') and 
'evidence' (E). It assigns an epistemic probability of P(HnE)IP(E) to the 

hypothesis after receipt of the 'evidence ·. and this formula gives the correct 
answer -so long as the vague Eis interpreted as referring to the second of the 
dual chance-processes distinguished above. To give the right ans wer, E has to 
be short for the sentence 'the evidence is acquired' -so that P(E) is the 
probability of acquiring the evidence (via that second process). 

So in our family example above, where boyness was only revealed if the 
family had two boys, P(E) can be interpreted as 'the probability of the 
observation-process indicating there is at least one boy in the family', and 
this is here k· So is P(HnE). In consequence, the revised probability recom-
mended by this version of the conditioning process is now 1. a satisfactory 
ans wer. The calculation simply repeats that already carried out using Y* above. 

In many other situations such an interpretation of E (as referring to the 
second process) comes naturally, and the formula is repeated1y applied so as to 
be identical to the rule I formulate using Y*. It works weil-and in sofaras it 
does so. it supports my Y*-rule. Indeed, it does this so weil that it might seem 

my Y*-rule adds nothing to it, so the whole discussion here is pointless, 
targeting a mere straw man. 

Butthis is not so. Firstly, and relatively tritely. because it is not realized that 
the successful H-E formulation is so different from the standard X-Y formula-
tion setout above. In particular, the successes ofthe H-E evidence-formulation 
are not deemed grounds to reject the X-Y outcome-formulation. In conse-
quence, and far more importantly, there are important cases in the Iiterature 
where this H/E-formulation is used with E referring to Y, the content of the 
evidence, as opposed to Y*. the fact that it is received. So even if it is true that E 
is sametim es interpreted correctly, it is not ahmys interpreted correctly, and 
there is real need to distinguish the Y and Y* formulations-to avoid a 
confusion which Korb ([ 1994], pp. 142-5) aptly calls ·propositional mesmer-

ization'. I gi ve two examples immediately below. both of which have rela-
tively serious consequences. beyond mere quantitative mis-estimation (and 
Korb.loc. cit., gives another). So the H/E-formulation is in fact ambiguous and 
invites both interpretations, the false and the correct. My aim is to eliminate 
this source of confusion, by forcing the two interpretations further apart. 

A potent example ofthe failures here is provided by a familiar application of 



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What Are Conditional Prohabilities Conditional Upon? 677 

Bayes' Theorem to situations in which somc prediction made by a scientific 

theory His observed. Howson and Urbach argue ([1993]. pp. l19ff) that the 
probability of the theory is routinely increased by such a confirmation (and 
many others agree, e.g. Glymour [ 1980]. p. 92 and Salmon [ 1967]. p. 117). 
But the analyses here are fallacious, because they misidentify the evidence E, 
taking it tobethat which is observed (my 'Y'), rather than thefact that it is 
observed (my 'Y*'). For Howson and Urbach takc P(EIH) tobe I (deducing 
this from the stipulation that H =* E) and hence guickly infer that the 
posterior probability of the theory, P(HIE) = P(H)IP(E). Thi~. of coursc, 
can never be Iess than the prior probability P(H). since P(E) :S I, and will 
greatly exceed P(H) whenever P(E) is small. So they conclude (op. cit .. 
p. 120): 'any evidence [ ... 1 must confirm every hypothesis [ ... ] ofwhich 
it is a logical conseguence' (so lang as we stay outside the territory of 
evidence and theories that are so implausible their probabilities vanish 
altogether). 

To display the fallacy here, Iet us Iook at the concrete example provided by 
our two-child family. We are asking how one should update thc probability of 
the hypothesis H (with prior probability ~) that the randomly chosen family has 
two boys, after obsening something that follows from the truth of the hypoth-
esis-that the family has at least one boy. 

Let E be the sentence expressing the observed conseguence of the hypoth-
esis, i.e. 'the family has at least one boy'. Then E has prior probability P(E) =; 
and H =* E. Let further E* be the fuller Statement of the 'evidence'. the 
sentence 'the family is observcd to have at least one boy'. A probability cannot 
be confidently assigned to E* without more information about the observing 
process; but it is certainly not true that H =* E*. 

Because E* is not cntailed by H. this cannot be what Howson and Urbach 
(and others who endorse this analysis) mean by the 'evidence'. They must 
mean something closcr to my E, and I must assume this is what do mean. as 
there are no plausible alternatives. Then indeed, the conditional P(EIH) 

becomes I (as they presume). 
Given this choice of evidence E, however, the posterior probability has tobe 

evaluated as *· since then P(HIE) = P(H)/P(E) = ~/ ~- Yet this is the answer 
already noted above as being grossly wrang. 

A better treatment would use E"' as the evidence. to yield the posterior 
probability 

P(HIE*) = P(E*IH).P(H)/P(E*J 

which can readily be less than the prior probability. That happens whenever 
P(E*IH) < P(E*). I.c. if some conseguence of a hypothesis is especially 
unlikely to be observed when the hypothesis is true, thcn witnessing that 
conseguence can decrease the plausibility of the hypothesis-as intuition 



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678 Keith Hutchis(m 

would surely suggest. 12 Howson and Urbach and their supporters g1ve an 
analysis which denies this, for they misrepresent the evidence, misled by the 
ambiguity of the Hf E-treatment of conditional probabilities. My target here is 
no straw man! 

We can see much the same phenomenon in a farnaus discussion of Jeffrey's 
(11983], p. 165 ). centred on the updating of the probability of a hypothcsis after 
arrival of evidence which is so vague that its content cannot be put into words. 
Jeffrey claims the Rule of Conditionalization does not apply here. because 
there is no proposition (which I denote 'E') encapsulating the content of the 

new information acquired. Yet (as Jeffrey hirnself notes. op. cit.. p. 166) there 
is a proposition (my 'E*') encapsulating the fact that thc ineffable evidence has 
arrived-or Jeffrey would not have been able to describe hisproblern to us. So 
the applicability of my proposed alternative rule is not called into question. 
This does not say that the revised formula is correct (or helpful), but it does 
confirm that it differs from a widely endorsed interpretation of the Standard 
Formulation ofthe process ofupdating. (In fact, thc revised rule seems to make 
Jeffrey's special rulc quite redundant. but I do not argue that case here.) 

We are finally ready to begin our main critique of that formulation. As 
already indicated. we use variants of a simple um experiment, designed to 
model the two-child family problem-but simplified and elaborated so as to 
emphasize the role of the observation-processes that provide the updating 
information. 

3 The negative argument 

3.1 The basic chance-process 
Let us imagine an um contains (say) eight coins, 13 two with two heads on them, 
another two with two tails, and the remaining four with one tail and one head. 
Suppose a totally honest scrutineer ('Eve', we will call her) causes the state of 
these coins to change in a seemingly 'non-deterministic' manner, by making a 
random draw of one coin from an um. 

le Bcfore I arrive at work, I am 20% confident that both secrctaries will be away (becausc of a 
severe epidemic of ftu). After I arrive at work I find a note telling methat one ofthe secretaries is 
away. But in consequence of finding this note, my con11dence that both secretaries are away 
drops dramatically-bccause by far the most plau>.ible source for the note is the other secretary. 

Let: H be the sentence 'both sccretaries are away': E bc the sentence 'at least one sccretary is 
away'; and E* be thc sentence 'I find a note telling methat at least onc secretary is away'. Then 
H => E. But the truth of E* (i.e. obscrvation of E = ohservation of a comequence of H) virtually 
guarantees that H is false. 

Foranother well-known example (due to Riebmond Thomason). see V an Fraassen ([l9R4], p. 
246). 

11 At this stagein our analysis the S hereisnot important. and any multiple of 4 (including 4 it>.e\0 
would serve equally well. The 8 is chosen as the smallest numher that will accommodate a 
variant of thc problem introduced below. 



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What Are Condirional Probabilities Conditional Upon? 679 

Suppose an observer ('Adam') is interested in betting on the head/tail 

properties of the coin chosen. For such an observer, the draw (our 'evolu-

tionary' chance-process. as articulated in Section 2.2 above) can be weil 

modelled by choosing a 3-member outcome space ü ~ { hh, ht. tt}, where 
'hlz' stands for the outcome-type description 'a two headed coin was chosen', 
and so on. Since the choice was declared to be random. the function p with 

values p( { hh}) = p( { tt}) = ~ & p( { ht} )= t. etc. will suffice to make the pair 
d2, p> a finite probability space (in the sense of Kolmogorov) that con·ectly 

models the chance-process. (I do not defend this claim, simply presuming the 

reader's familiarity with such matters.) 

Let H be the set { hh}, representing a draw that produces two heads. and Iet 
us consider at length the way our observer Adam should update the probability 

he assigns to this two-head possibility as his epistemic circumstances change. 

If all Adam knows is that implicit in the specification of the initial draw, he 

should assign a prior probability to H of p( { hh}) == ~-
Suppose Adam Iater participates in some observation-process, and thus 

receives additional information about the head/tail-state of the drawn coin, 

indeed that it has at least one head on it, i.e. that an outc0111e in the subset E = 
{hh, ht) has been achieved. (Note that p(E) == ~.) If it is also true that this 

constitutes the totality of his new knowledge. then the traditional formulation 

of the Rule of Conditionalization is paradigmatically applicable. For we have 

all the prerequisites for application of that Rule in place: an evolutionary 

chance-process and an observational chance-process; an acceptable probabil-

ity model for the evolution-process; and finally, the observation-process has 

revealed that an event represented by a subset of the outcome space has 

occurred. According to the Standard Rule, this is all we need to know to 
revise the probabi lities the observer should assign to all other outcomes-and it 

teils us that the new probability tobe assigned to His unequivocally *· 14 
For reasons that will be spelled out at length soon. I do not accept this 

answer; and I suspect that many readers (those who reject the analogous 

answers to the family-with-one-boy problem) already agree with me. For the 

moment however. that is not what matters. The important point is to accept that 

one is committed to this answer if one endorses a version of the Rule of 

Conditionalization Iike that articulated in the opening of this paper. 

A defender of the Standard Rule could, it is true, deny this by insisting it is 

impossible to find out that the outcome is of type E without finding out 

something eise. Indeed. every time one discovers something about the coin 

drawn, one will also learn something about oneself. e.g. that one's eyes have 

recently been open. So E will never represent the total evidence here, and the 

14 Forthat Rule assigns a posterior probability given by thc fonnula P(H nEi/P(EI which now 
take' the values p( (hh} n( hh, ht })lp( ( hh. ht}) = p( ( hh} )/~ = Yi = ~-



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680 Keith Hutchisan 

conditions for application of the Rule arenot met. I do not have any compelling 
response to this version of the objection. except to remark that it obviously 
undermines many other versions of the Rule of Conditionalization. If this 
objection were taken too seriously. it might weil protect such rules from 
refutation, but at the dramatic price of proteering them from application. No 
sentence could ever be formulated to encapsulatc the full evidence, and every 
case of conditioning would be confronted by Jeffrey's difficulty (noted in 
Section 2.3). 

One must (I presume) deem that this particular piece of additional knowl-
edge is irrelevant. But as indicated above. I have no useful account to offer of 
how we decide what is relevant and irrelevant. and hope that it is relatively 
easy to make this decision in individual cases. In the end. we will observe that 
relevance is not the big issue, so I am in no hurry to deal with this problem! 

3.2 Elaboration of the chance-process 
A far more serious version of the same objection observes that detection of an 
outcome of the evolution-process always brings with it information about 
some details of the observation-process. One cannot find out that the coin 
drawn has at least one head on it without understanding something about how 
one found out the coin possessed that head. That 'something' must be accom-
modated when assessing the total evidence, unless we deem it irrelevant. 
following the strategy above. But that exit is rightly closed to us here. since 
an insistence upon the relevance of the observation-process constitutes the 
germ of my case! So the objection is a valid one. 

Yet its impact is slight. For it remains true that the conventional formulation 
of the Rule of Conditionalization does not force us to accommodate the source 
of our new knowledge. To see this, Iet us consider a slight modification of the 
above chance-process, one designed to leave the evolution-process unaltered 
in all essentials, but with details of the observation-process spelled out in 
advance. In consequence (as my concrete example should make clear) the 
obscrver Adam's understanding of the observation-process is acquired sepa-
rately from (and well before) the new knowledge about the outcome of the 
evolution-process-hence cannot now get included in the 'total evidence'. 

So Iet us now suppose that (in this revised process) our scrutineer Eve takes 
the coin behind a screen after making her draw. There she makes several 
decisions, via processes declared in advance and understood by Adam. After 
implementing these decisions, she emerges from the screen with the selected 
coin flat on a saucer but covered. She then shows the saucer to Adam, and 
removes the cover. His eyes then see the coin. quickly detecting whether it is a 

head or a tail that is on show. 
Prior to Adam· s eyes seeing the coin, the probability of there being a 



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What Are Conditional Prohabilities Conditional Upon? 681 

two-headed coin on the saucer is 1· How should this probability be modified 
just after his eyes idcntify a head? 

The standard version of the Rule of Conditionalization still applies to this 

revised chance-proccss, and it commits us to the same answer as before, *· For 
an observer interested only in heads and tails can still model the elaborated 
process by exactly the same probability space as was used to model the simpler 
process, for the two evolution-processes are effectively identical: they end up 
with the same three possibilities for the head/tail-state of the coin drawn: and 
the prior probabilities of these statcs are quite unchanged. 

In this case, howevcr. it is implausible to insist that additional knowledge of 
relevance was acquired at thc time the head was recognized: for Adam 
possessed a full understanding of how he was going to find out about the 
coin before his eyes made the identification. The only things he ascertains as a 
result of identifying the coin face is the fact that the coin has at least one head 
on it-that the outcome is in the subset {lzh, ht). This knowledge 
now constitutes the total new evidence of relevance. So thc Standard Formula-
tion of the Rule does indeed apply. and it Ieads to the posterior probability I 

reject, namely ~-

3.3 Refuting the Standard Formulation 
The reason that this posterior probability * is unsatisfactory. is that the prob-
ability here obviously depends on what decisions the scrutineer Eve imple-
ments behind the screen. 15 If she had agreed. for example, to show a tail 
whenever possible, then she will only show a head when the coin has two 
heads-and Adam (we have agreed) will know this. So the posterior prob-
abilitytobe allocated by a wise observer to the coin's having two heads when 
one head has been revealed is l. Two heads now becomes a certainty after 
seeing one head. 

Not only does this clearly correct answer differ from that provided by 
the Standard Formula, 16 but more information has been used in its 

1
' This fact is apparently welJ recognised by many statisticiam •. though secms oddly obscured in 

their discu"ions. Sec c.g. Grimmet! and Stirzaker ([ 1993]. p. 24. problem 26), but note that the 
phcnomenon is hidden in a problem. while an indcx search for thc critical word 'protocol" Ieads 
to no exposition of note. For some other examples of the recognition. see the citations in fn. 8 
abovc. 

1° From oral discussion it has become clear that this claim is controversial. Yet those who dispule it 
are often unconsciously correcting the Standard Formulation in exactly the direction I recom-
mend. The critical issue ;,, what onc takes as the denominator to evaluate the posterior 
probability, the prior probability of getting at least one head. or the prior probability that one 
will find out that one has got a head. Thc formula erroneously takes the forrner option. for it says 
that one uses P(Y) the prohability of getting (in the cvolution-procc". the one generating the 
priors) the discovcred outcorne Y. This Y is what the observer finds out. so P( Y) is not the 
probahility of discovering Y. An event might have a high probability of taking place. but 
discovery of the evcnt might be very unlikcly. 



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682 Keith Hutchisan 

evaluation than the Standard Formularion deems to be required-more has 
bcen used than just the values of P(XnY) and P(Y). Indeed, knowledge 
restricted to the outcome-probabilities of the evolution-process-the 
simple selection of the coin-does not here enable Adam to update the 

probabilities. He cannot reach the answer of the last paragraph using such 
information alone. for his understanding of the observation-process was 
clearly vital there. (This is why I earlier described the Standard Formulation 
as pretending to usc too littlc data, and hcncc as 'strongcr' than thc process I 
articulate.) 

The answer provided by the Standard Formulation (vi;:;. *) does. however, 
apply when an appropriate observation-process is involved. If indeed Eve 
agrees to show a head whenever possible, the wise observer who understands 

that this is how she is deciding, should now allocate * as the posterior 
probability. (For a head will then be revealed after ~ of a-11 draws in a long 
run; but two heads will be present in! of draws. all ofthem included within the 
where the head is revealed. So the two-heads cases will constitute * of the cases 
where a head is revealed.) 

Yet if Eve simply tosses the coin, and reveals the face which comesdown 
uppermost, the situation is more complicated, for the bias on the coins becomes 
important-even though that (obviously!) had no bearing on the outcome 
probabilities of the evolution-process, the draw from the um. A fortiori, the 
bias cannot be reftected in the any conditional probabilities calculated using 
the Standard Formula. But they can be accommodated in more elaborate 
calculations that attend to the observation-process. 

lndeed, if the coins are known tobe unbiased. the posterior probability (for 
an observer who knows Eve makes her decision by tossing the coin) is easily 
evaluated as !· But if the coins are biased (and perhaps differcntly biased). a 
whole range of posterior probabilities can be provided. according to the 
degrees of bias. If the bias is unknown, I do not know what to do-but sec 
Section 4.3 below. Yet if the observer knows the bias, the calculation is 
straightforward: he uses knowledge of the bias. to calculate how often a 
head will be seen in a long run; he then calculates in what proportion of 
those cases, two heads will in fact be on the coin-and takes that proportion to 
be the posterior probability sought. 

So after the outcome Y of some evolution-process has been detected, 
the posterior probability to be allocated an outcome X depends on more 
than X, Y, and the prior probabilities of the outcomes ofthat process. lt varies 
with the epistemic procedure as weil. That establishes sub-thesis (e) in my 
summary above, and constitutes the core of my case. It also explains the title 
of my essay: for as the table below indicates in summary, conditional 
probabilities are conditional upon the protocols governing the release of 

information. 



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What Are Conditional Prohabilities Conditional Upon? 683 

Table 1. 

Protocol Used by Honest Scrutineer 
to Reveal at Least ONE Head 
on Coin Drawn 

Reveal head only if impossible to reveal tail 
Toss (unbiased) coin, then reveal upper face 
Reveal head whenever possible 
Reveal head using complex protocol in 

Section 4.7 
Reveal head only if coin has head and tail 

Probability to he Accorded to 
nvo Heads. Ajier Discm·ering 
at Least ONE Head 

100% 
50'!c 
33.3% 
20'/c 

0% 

4 The positive supplement 

4.1 Correcting the Standard Formulation 
The Standard Formularion of the Rule of Conditionalization claims that after 

ascertaining that an outcome of type Y has been achieved, one can calculate the 
posterior probability to be accorded to some other type X. using nothing 
beyond the range of prior probabilities of all outcomes of the evolution-
process. The example just given shows that this is not true, yet makes it 
fairly obvious how to correct the formula more generally. Clearly, one 
cannot update the probability without access to the probabilities attached to 
the observation-processes that provide the updating information about 
the outcome achieved in the evolution-process under Observation. lndeed, 
the two critical probabilities are: (a) the probability that one will receive the 
evidence in question. P(Y*) as I have called it above; and (b) the probability 
that BOTH some outcome of type X will have been achieved in the process, 
AND the updating infonnation will have been received by the observer, i.e. 

P(XnY*). The posterior probabilitytobe assigned to Xis then just the ratio of 
these two quantities, P(XnY*)IP(Y*). 17 This is the replacement formula urged 

in the operring of the paper. 

4.2 When does the Standard Formulation work? 
Despite this corrcction, there remain circumstances in which the Standard 
Formularion gives an acceptable answer. and it is worth briefly noting some 

17 Exactly why this revised formula is true is hard to tcll, and thc problem of dcfending it is just a 
variant of the famous problern of justifying the convcntional formulation. Sec e.g. Howson and 
Urhach ([1993], pp. 99ft); Mahcr ([1993], pp. 120ft). A frcquency analysis is clear and works 
weil in situations (likc my urn draws) where frequcncies serve as a reliable guide to probabilities. 
I do not attempt a more elaborate justification, as I doubtthat will be an issue for many readers. 
They will bc unlikely to dispule the rule. 111erely claiming that it is what ü, already intended by 
thc Standard Formulation. Hencc the detour to confront this question in Section 2.3. 



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684 Keith Hutchis(m 

examples of these. The obvious sufficient (but certainly not necessary: sec 

below) condition for this, isthat the observation-process besuch as will ensure 
that both P(}) = P(Y*) and P(XnY) = P(XnY*). These two requirements 
furthermore will bc met (trivially) whenever the observation-process is such 
as might weil be called 'persistent' -meaning that it never fail to in form ofthe 
truth of Y if Y is indeed true. So persistence is a sufficient condition for the 
Standard Formulation-granted, of course. that the observation is also ·reli-
able · (in the sense of never indicating Y to be true when it is not). Such 
·reliability' is implied by the presumption we have been making (right from the 
beginning of our discussion) that our observer is acquiring knowledge, but 
persistence (in my technical sense) is an additional requirement, and it is 
routine for epistemic process nottobe persistent. Many things are true without 

our finding out that they are true 1 

If we think of the observation-process as a test (for Y-ness, of course, not 
X-ness !). then this condition for the accuracy of the Standard Formula is that 
often expressed by saying that the test produce neither false negatives (my 
·persistence') nor false positives ('reliability'). Under such conditions, the 

Standard Formularion will clearly provide the right answer. 

So in the um draw example, we agreed with the answer *· if it was true that 
the scrutineer had agreed to reveal a head whenever possible, i.e. 'persistently'. 
Andin the two-boy family problem. the updated probability * seems correct, if 
the observer knows the family was randomly chosen, and that he has found out 
that it has at least one boy in it by some persistent procedurc, one revealing the 

existence of a boy whenever a boy is there. Being told that thcrc is a boy in the 
family only when a girl cannot be revealed, is not such a procedure. That is why 

the updated probability in these circumstances ( viz. I) is not the * provided by 
the formula. 

Thesefacts have a further consequence of importancc. for they show that the 
principle oftotal relevant evidence (as embedded in the Standard Formularion 
of the Rule of Conditionalization) is false: new knowledge does not have tobe 
restricted tothat allowed by this principle; and imposition of such a restriction 

can generate inaccurate updatings. For we have seen that one can confidently 
assign the posterior probability P(XI Y) to X, after ascertaining Y, if one has 
additionally ascertained that the information about Y was provided a persistent 
and reliable procedure-and if (as weil) nothing eise of relevance has been 
ascertained. There is clearly no necessity that the knowledge of the epistemic 
procedure here be acquired in advance of the knowledge of Y, as the timing of 
its arrival is not vital. So it is quite reasonable for this knowledge about the 
procedure to arrive slightly after the knowlcdge about the outcome of 
the evolution process. It would then be part of the new knowledge that creates 

the need to update probabilities. 
New knowledge does not then have tobe as restricted as the principle oftotal 



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What Are Conditional Prohahilities Conditional Upon? 685 

evidence often declares. To assign the posterior probability P(XIY) to 
each outcome of type X, after ascertaining that an outcome of type Y has 
been achieved, it is not necessary for the knowledgc that Y has bcen 
achieved to represcnt thc totalitv of relevant new knowledge acquired. 
Highly relevant knowledge-knowledge that the epistemic proccss was 
persistent can also be acquired-without jeopardizing the assigncd posterior 
probability. 

To relax the rule of total evidence to accommodate this fact is, however, 
relatively easy. and involvcs a mere rewrite of our earlicr Rule of Conditio-
nalization. Doing this then gives us a restricted version of the Standard 

Formulation, one that now seems to give acceptable posterior probabilities, 
albeit within its limited scope. Using italics to indicate the modification, that 
revision reads: 

Suppose circumstances aresuch as enablc the reasonable ohscrver of some 
chance-process to place contidence P(X) in the outcome hcing of type X. 
THEN P(XnY)! P( Y) represcnts the confidence that obsen·er should place 
in the outcome heing of type X after the circumstances of thc ohserver 
change, through addition ()fjust TWO relevant pieces of knowledge-l'i::. 
that some outcome of r.vpe Y has in jäct heen achieved. AND that this 
indicatimz o{ Y -ness was provided bv a 'persistent· epistemic process. 

Conversely. if newly arrived knowledge is morc restricted-in accordance 
with the standard version of the principle of total evidence-to knowledge 
of the outcome-type Y alone. one cannot confidently update the probabilities 
of the other outcomcs. For if one does not already know thc procedures, it 
is not clear how to update the probabilities. lt does remains clcar, 
however, that one cannot trust the unadjusted formulation of the Rule of 

Conditionalization. 
But persistence is a rather strong requirement, and incompatible with most 

everyday epistemic activity (though quite acceptable in the careful model of 
conditioning that I sct out in Section 4.5). Reliability is (perhaps?) more 
common, and when information is received reliably, the whole discussion 
above can be significantly broadened. To illustrate this. I change temporarily to 
the language of 'hypothesis' and 'evidence·. as this secms to make our new 
situation come alive. For the Standard Rule also gives a correct updating when 
thc truth of some hypothesis H makes no difference to the transmission (as 
opposed to generation) of a piece of evidence. For P(HIE) then has the same 
value as P(HIE*), as can quickly be deduccd from our assumption that 
P(E*IE) = P(E*IHnE), since reliahility ensures that both P(E*nE) = P(E) 
and P(E*nH) = P(E*nE nH). But we still necd to know these facts ahout our 
epistemic activity to apply that Rule, so again thc principle of total evidcnce 
has to be relaxed. To update successfully, we need to have grasped our 

epistemic history. 



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686 Keith Hutchis(i/l 

4.3 Updating in ignorance of the epistemic procedure 
To see this last point vividly, we need only consider a variant of the chance-
process last set out, whcre the scrutineer Eve remains honest (so that all 
information released is 'reliable' in the sense just noted). but where the 
observer Adam is now denied knowledge of the decision procedures used by 
Eve, and where. furthermore. she is not obliged to release information after 
every draw. 

Every time his cyes identify a head, the observer Adam will still know with 
certainty that the coin drawn possesses at least one head, so continues to 
acquire knowledge about the outcome of the cvolution-process, as presumed 
in the unadjusted Standard Formulation. But he will not now be in a position to 
assess the weight of his evidence. Eve may. for instance, be refusing torelease 
information when there are two heads on the coin drawn, so that it is impossible 
for there tobe two heads if one head is seen. Conversely, Eve may be doing the 
cxact opposite. showing the head only when there are two heads on the coin: 
the coin then has to havc two heads if the observer sees one head. But Adam 

cannot distinguish these two diametrically opposed situations. He is in con-
sequence dramatically hampered in allocating posterior probabilities, and can 
have no confidence in the answer provided by the Standard Formulation. 

Adam is not completely incapacitated by these facts, however, and can 

certainly make guesses based on all sorts of evidence about how he believes 
Eve is making her dccisions, and hence estimate the critical prior probabilities, 
a: [== P(Y*). the probability of his having acquired the knowledge that an 
outcome of type Y has occurred] and ß [== P(X nY*). the probability of 
BOTH his having acquired this knowledge AND an outcome of type X 
having occurrcd]. But the posterior probability he will thus assign to two 
heads ( viz. ß!o:) is no more reliable than these guesses. and can be badly 
discordant with the reality of the situation. I.e. if the scrutineer Eve is choosing 
by a procedure totally differenttothat the observer Adam believes she is using. 
the long-term frequency with which two heads occurs among situations where 
the single head is revealed can be dramatically different to the probability the 
observer subjectively holds. Just how entitled he is to hold this probability in 
such circumstances is a question I do not wish to pronounce upon; 18 but we 

'" A subjectivist may im. ist tbat since probability represcnts a dcgree of belief. the observer has no 
choice but to hold the 'unsound" probability. But such a subjectivist will have to deny the 
standard Kolmogorov axiomatization, which attributes a probability of I to every tautology. For 
there are plenty of tautologies so complicated that somc ohservers will not bclieve they are 
tautologies. 

I do not know how to resolve this dilemma. The positive case I present here seems as strong 
(and weak) as the claim thal a tautology has tobe give thc prohability I. My negative case is far 
stronger. for it is clear thc Standard Formulation does worse. lt will not even allow the observer 
to hold the conditional probability allocated in the light ofhis suspicions as to how information is 
being provided. It insists uniformly on the posterior prohahility associated with a persistent 
observation-proccss. 



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What Are Conditional Probahilities Conditional Upon? 687 

must certainly recognize that the probabilitywill fail as a guide to safe betting 
ratios. It is (in some serious sense) a defective estimate of the posterior 
probability. To update probabilities safely, onc needs access to the something 
beyond that allowed by the unadjusted rule of total evidence~the probabili-
ties governing the release of information. 

4.4 Updating upon receipt of irrelevant or incorrect 
information 

The whole of the discussion above has emphasized the importance of distin-
guishing between Yand Y*, Ybeing the knowledge provided to the observer by 
the action Y*. Wehave observed as wellthat posterior probabilities are directly 
dependent on thc probabilities associated with Y* rather than Y. 

In consequence it can readily happen that it is wise to update probabilities in 
situations where the Y is irrelevant, so lang as the associated Y* is not 
irrelevant. What the observer finds out may weil be of no interest; but the 
fact that he 'hears about if can be highly significant. Indeed, it can also be wise 
to update probabilities upon receipt of information Y that is badly wrong 
(whether suspected to be wrang or not); and it is possible to perform this 
updating rationally if one knows enough about thc proccss that provided the 
dubious information. The principles involved in supporting these claims are 
quite straightforward. and an example should make them clear. 

We peruse indeed a further adjustment of the basic um draw above, attend-
ing now to the chemical composition of our eight coins. Suppose that four of 
the coins happentobe made ofbronze (with one bronze coin having two heads; 
one having two tails: and two having one head and one tail). Suppose also that 
the remaining four coins are made of nickel. Our obscrver Adam is to remain 
interested in betting on the head/tail properlies of a randomly drawn coin, and 
has no interest in the coins' chemical compositions. It is (I take it) obvious that 
the composition of the coin drawn has no bearing at all on its head/tail-state: 
the head/tail-statc of the coin drawn is 'statistically independent' (to use some 

common jargon) of its chemical-state. The probability of the selected coin 
having two heads on it is unatfected by the composition of the coin. and if one 
finds out that the coin is made of bronze one has found out something that 
would normally be deemed irrelevant to its head/tail-state. 

Butthis does not mean that one should avoid revising the probability ofthere 
being two heads on the coin after receiving the information that the coin is 
bronze. 19 For again. we must Iook at the observation-process. the rules 
governing Eve · s relcasing information. It could happen. for instance, that 

19 I.e. two outcomes can be stati,tically indcpendent in the ontic sense. yet epistemically depen-
dent. The persistent blurring of this distinction in the Iiterature (sec fn. 9 above) is simply a 
variant ver.sion of the problern attacked in the present paper ria the Rule of Conditionalization. 



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688 Keith Hutchisan 

the scrutineer has promised only to releases the information that the coin is 
bronze when it BOTH has two heads on it AND is bronze. Otherwise, she will 
release no information about the chemical composition of the coin. Instead 
she will then indicate that the coin has a tail on it (if the coin does not have 
two heads), or that it has a head on it (if it has two heads but is not bronze). 
Undersuch conditions, the rational observer who knows in advance that this 
is how information is going tobe released will assign a posterior probability 
of 1 to the coin' s having two heads after discovery that it is bronze. 
Obviously, such a posterior probability is again provided by the formula I 
recommend, P(XnY*)IP(Y*) (for XnY* = Y* here). Yet the traditional for-
mulation Ieads to no change in the probability after discovery that the coin is 

bronze. 20 

It should also be obvious now that Eve's honesty is not an issue either. It is 
her practices that matter. not the words she uses to describe those practices, and 
discord between the words and the actions is of no interest to any observer who 

understands the actions. 21 Suppose indeed that Eve claims to release informa-
tion as outlined in the last paragraph, but in fact does something different, 
revealing a bronze composition only when there is a head and a tail. The 
observer who knows this should simply ignore what Eve claimstobe doing. So 
if bronze is revealed to him, he will assign a posterior probability of 0 to the 
coin · s possessing two heads. And again, such a posterior probability is recom-
mended by the formula I propose, for now P(X nY*)/P(Y*) = 0 since in these 

circumstances P(X nY*) = 0. 
Suppose, indeed. that the scrutineer were to make her 'dishonesty' patent, by 

declaring a familiar falsehood (e.g. that 2 plus 2 is 5) every time there were two 
heads, but revealed a chemical composition in all other circumstances. The 
observer who understands her actions is still able to update the probability of 
two heads. despite the gross irrelevance of her false claims. What matters to 
him is not the overt information declared in the scrutineer's message, but what 
might be called the 'latent' information, what one can infer from the fact that 

the claim has been made. 

4.5 A revised model and formula 
Given then that the overt content of the message received by the observer is 
irrelevant to the transformation of prior probabilities into posterior ones, it 
seems a good idea to reconstrue our paradigm cases of conditioning, by 

20 The probability of getting two heads on a bronze coin is 1/8; thc probability of getting bronze is 
1/2. The conditional probability of gctting two heads given a bronze coin is just the ratio ofthese 
two probabilities, i.e. 1/4. This is the same as the prior probability of gctting two heads. 

'
1 An observer who does not understand the actions may of course use the words as a guidc to the 

scrutineer's actions. and then honesty. etc. does become important. 



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What Are Conditional Probabilities Conditional Upon? 689 

imagining that the messages received are particularly bland. This reduces the 
risk of our being side-tracked by red herrings. So Iet us suppose that the 
information provided to the observer (in the observation process) takes an 
extremely simple form, say the illumination of a lamp, that comes on in one of 
two colours, green or red. This is intended to provide us with a relatively 
abstract model of a typical 'test' -to which there are three standard reponses: 
'still waiting to hear' (= no light); 'the testwas positive' (= green light); 'the 
testwas negative' (=red light). 

More fully, I suggest the following type-specimen of the problern of con-
ditioning. A chance-process takes place. A rational observer of the process 
does not witness its actual outcome, but does attribute probability P(X) to each 
possible outcome X. After the process has completed. the observer sees a light 
come on and display the colour green. He receives no other signals of 
significance. Should he revise the probability he assigns to each outcome X 
after seeing the green. and if so, how? 

Before answering this question. we should briefly note that no generality is 
lost here if we presume the observer is behaving in such a way as to recognize 
the green light every time it comes on, i.e. that the information process is 
persistent in the sense described above. 22 Realizing this enables us to simplify 
greatly the discussion below, since it means that there is no need to distinguish 
between epistemic and ontic conditionals. For the sentence 'the green light is 
on' is true in exactly the same circumstances as the sentence 'the observer 
realizes that a green light is being displayed'. 

It is, of course, obvious that there are circumstances in which the observer 
should update probabilities after seeing the green signal. It is much less 
obvious whether there is any general algorithm that governs this updating. A 
typical situation in which it is desirable to update is when there is a causallink 
between the unwitnessed outcome of the evolution-proccss and the colour of 
the light displaycd. The critical factor is the degree of correlation (however 

22 For Iet us suppose this were not so. that the observer was watehing a light that shows eilher blue 
or yellow, and that he only saw the yellow interrnittently. To conditionalize on the non-persistent 
yellow signal is. however, the same as to conditionalize on an imaginary but persistent green 
signal that comes on whenever the observer recognizes yellow. So once we invoke the 
abstraction here of using a simple coloured-light-signal as a standardized representation of 
the conditionalizing inforrnation, there is no need to allow for non-persistent signals. Each of 
them can be replaccd by a persistent one. 

ln the language of tests, we are presuming that the observer always linds out the result of the 
test, on thc grounds that a tcst whose results only reach the observcr intermittently can be 
modelled by a different test. one whosc rcsults always reach the observer. The modeHing test is 
that variant test which is positive if BOTH the modelled test is positive AND the observer linds 
this out-negative otherwise. 

We arenot prcsuming here that the modelled test itself nevcr produces false negatives. That 
would require us to make presumptions ahout more than just how often the green light is secn. 
We would need to presume that the greenlight came on every time the property it was intended 
to revcal was present. We cannot judgc whether a test produce-; false negatives (or positives) 
without knowing the purpose of the test. lt is not a pruperty of the test simpliciter. 



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690 Keith Hutchis(J/l 

generated) between each outcome X ofthe evolution-process and the display of 
a green light. hut to say that is to do little morc than rephrase the question 
asked. 

But at least that rephrasing enahles us to focus on what matters, and it is easy 
to analyse these in seemingly unproblematic frequentist terms. How often does 

the green light come on? In how many of those cases is X true? The updated 
probability we seek is surely-but see fn. 17-just the ratio of these two 
quantities. i.e. P(X is true & light is green)/ ?(light is green). 

4.6 Abbreviating the revised model 
This 'ans wer' can be expressed rather differently. We can stop thinking of 
conditioning as involving a pair of chance-processes (one involving the 
temporal evolution of the state of the primary system, that under Observation, 
and the other involving the temporal evolution of the state of our signal-
light). And we can start thinking in terms of a single, but two-stage, chance-
process, a single non-deterministic evolution occurring within a composite 
physical system. The new evolving system isthat composed of the primary 
physical system under observation plus the signal-light. The two-stage 
evolution of the composite system produces outcomes which are pairs of 
the form (X,S) meaning that an outcome ofthe type X occurred in the primary 
system, and this was followed by the green signal. lt would be natural (and 

formally possible) to refer to this pair using a notation like XnS; and 
similarly to use S to refer to the composite outcome (!.1, S)-where !.1 is the 
set of all outcomes of the primary system. The symbol S then describes the 
event 'the signallight came on', irrespective of what happened in the primary 

system. 
Once that is done, the formula for the posterior probability simplifies. Now 

invoking nothing but probabilities of outcomes of a single (albeit composite) 
chance-process, it declares the posterior probahility to be the familiar ontic 
conditional ?(XIS)= P(XnS)IP(S). 

Of course. this Iooks just like the formula in the version of the Rule of 
Conditionalization that has been the target of this whole discussion. But the 
similarity is merely superficial, for the symbols have quite different meanings. 
The big difference isthat my formula incorporates the probabilities associated 
with the observer's receipt ofthe information, while the Standard Formulation 
ignores this. The latter's denominator P(Y) is simply the probability of the 
outcome Y being achieved in the primary sub-process within our composite. lt 
makes no essential reference to the means by which information about that 
outcomes is transmitted. The denominator in my formula by cantrast makes no 
reference to the outcomes of the initial process. It refers exclusively to the 
behaviour of the signalling process. 



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What Are Cunditional Prohabilities Conditional Upun? 691 

4.7 An illustrative example 
The discussion of thc last two sections was inconveniently abstract. and it is a 
good idea to see how it applies to concrete situations. So Iet us elaborate our 
discussion: by (firstly, in the prescnt section) observing the signal-light ana-

lysis avoids the defective answer (*) to the two-boy family problem: and 
( secondly, in the next section) by using the abbreviated model alone to system-
atically evaluate one of the puzzling probabilities associated with a traditional 
paradox. 

The observational-process we imagine in our first illustration is going tobe 
convoluted, perhaps inconveniently so-but there are sevcral good reasons for 
tolerating the inconvenience: firstly, because the convolutions model the fact 
that information-ftow is often complex, with information sometimes withheld, 

and somctimes released. yet for different reasons in different circumstances, 
some of them erratic. and somc of them systematic or deliberate; and secondly, 
to demoostrate the ease with which the 'signal-light modelling' can handle the 

computation. Artificial simplicity would obscure that fact. 
So Iet us suppose that, after selecting the two-child family. a fair coin is 

tossed by the scrutineer. If it comes down heads. the scrutineer inspects the 
family. If it comes down tails. nothing eise happens and no light comes on. 
When the inspection is carried out, the scrutineer tosses the coin again if the 
family is found to contain two boys, and turns thc green light on, if she then 
gets a head.lfhowever, the family only contains one boy. she also turns on the 
green light. lf finally, the inspected family contains no boys, she turns the red 

light on. 
Prior to seeing the signal-light, a rational observer would assigns a prob-

ability of 1 to there being two boys in the family. Suppose this observer sees the 
green-light. but gets no other relevant information. beyond the details above as 
to how the light is operated. What updated probability should he then assign to 
there being two boys in the family'? 

To answer this question now, we only need to ascertain two things. The first 
of them is what I have designated P(S) in Section 4.6 abO\·e. thc probability the 
green signal-light comes on. This is just ~-<H + ~.1 + ±.0) = ~-(k + ~) = ~. ~ = fi;. 

The second of them is P(XnS), the probability that BOTH there are two 
boys in the family [ 'X'l AND the green signal-light comes on. This is just 
I I I I 
2·4·2- 16· 

The posterior probability is then P(X nS)IP(S) = tG-;- fi; = k = 20%. 
In this case, the observer knows for certain that the family contains at least 

one boy if the green light comes on, but his newly acquired knowledge of the 
family is no morc extensive than this. So if the scrutineer had refused to tel! 
him the details behind the operation of the light ( only assuring him that 
whenever the light was green there would definitely be a boy in the family). 



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692 Keith Hutchisan 

thc conditions for the application ofthe Standard Formularion would be met. If 

that formulation wcrc valid, the postcrior probability would be *· 
So an observer who believed the Standard Formulation could be badly 

exploited. The ~crutineer could, for example, offer to pay him $4 every time 
the family was found to contain two boys after the greenlight came on. Forthis 
privilege, the observer would pay the scrutineer $1 every time the green light 
came on. The observer would think he was sure to win. since for him (believing 
the Standard Formulation) the fair pay-out would be just below S3.34. But he 
would lose money fast, for he would only win on 20% of bets. He would thus 

get back an average $4 of every $5 he handed over. Yet it would never be true 
that he had been deceived by the scrutineer. and he would find no evidence 
(beyond his obvious losses) to support such an hypothesis: the family would 
truly be found to contain one boy every time shc tumed the green light on. 

So the two mies are very different. The observer should refuse to play unless 
told how rhe scrutineer is deciding to turn on the green light. The Standard 
Formularion is seriously misleading about rhe fair bets. That is rhe core 
message of all the discussion abovc. 

5 Some traditional-information puzzles: Monty Hall 
A number of traditional problems in probability (like 'Berrrand's Box Para-
dox·. The Two-Aces Puzzle'. or 'The 3 Prisoners Puzzle'. etc.) 23 derive their 
impacr from Korh's 'propositional mesmerization· (p. 12). and the associated 
confusion (exrricatcd above) between what I have designared Y and Y*. They 
baffie us by displaying a conftict hetween several plausible revisions of a 
prior probability afrer new information is reccived. Yet once one accepts 
thar a knowledge of what is discovered ( Y) is rarely cnough ro allow a 
sound revision-and then pursues the missing Y*-much of the puzzlemenr 
disappears. and rhe principles above provide a quick (and seemingly algorith-
mic) merhod of effecting the revisions. a method which enables us to avoid 
altogerher the agonizing ad hoc manoeuvres (of uncertain validity) that plague 

discussion of these problems. 
To illustrate this. Iet us conclude with a brief analysis of one of the 

traditional paradoxes, 'The Monry Hall Problem': 

23 For the box prohlem. 'ee Kyburg (I 1970]. pp. 34-5). plu' Schlesinger ([ 1991]. pp. 188-901. For 
the two-aces puzzle. 'ee Freund ([ 1965]. pp. 29. 44 ). plus (iamov and Stern ([ 1958]. pp. 37-42). 
For the puzzle of3 prisoner's (nm 'the ]2] prisoner's dilemma· 1 ). see Schlesinger([ 1991]. pp. 8-
11 1. plus JciTrey ( [ 1988]. pp. 226-8). For an cxtended discussion of the 'Monty Hall' dehate. see 
Vos Savant and Granberg ([1997]. pp. 5-17. 169-96). plus (for a discussion that contrasts 
sharply with mine) Howson and Urbach ([1993J), pp. 384-5. where the relevance of the 
compere's protocol is effectively denicd. Discmsions ofthe similar 'two-child family problem·. 
havc been cited freyuently above. especiallly in fns 5. 6 and 7. Por further citations of 
discussions of analogous prohlems. see Shafer ([ 1985]. p. 2641. For the yuite different 2-prisoner 
dilcmma. see e.g. Jeffrey (] 1983], pp. 15-201. 



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What Are Conditional Probabilities Conditional Upon? 693 

A contestant in agame show ('Mary') is shown three opaque doors in a 
wall, one white. one black, and one red. She is told by the comperc 
("Monty') that a valuable prize has been placed (at random. for simplicity) 
behind one of these doors, but a worthless picturc of a goat is behind each 
of the other two. and that she will get the prize if she can choosc which 
door conceals it. Mary nominales the white door as her choice. to which 
Monty responds by opening the red door, revcaling a goat. He then 
offers Mary the opportunity of switching her choice to black-in the 
light of the new information he has apparently just providcd. What should 
Mary do? 

To answer this question systematically. it is enough to produce a convincing 
evaluation ofthe probability Mary should allocate to the sentence W ( = 'The 
prize is bchind the white door)', after receipt of the new information provided 
by Monty. lf this is greater than ~. she should avoid changing to the black door. 
If it is equal to 1. she ncither gain-s nor loses by the change. If it is less than !- she 
should change. In the light of the discussion above. the probability we need to 
evaluate is just the ontic conditional P(WIR*) where R* is the sentence 'Monty 
reveals a goat behind the red door'. So we can readily evaluate the probability 
we seck using the Standard Formula P(WnR*)/P(R*). 

Yet to evaluate each of P(WnR*) and P(R*) we clearly need to penetrate 
Monty's psyche-a feature ofthe problernthat sometimes seems puzzling. So 
IF Monty has just chosen randomly between the doors not selected by Mary 
(and thus run the risk of revealing the prize to her), then P(R*) = ~. ~ + 1.0 = *; 
while P(WnR*) = ~- so the posterior probability becomes ~- and the new 
information should not induce Mary to change. (But to realize this herself. 
Mary of course needs to know how Monty made his choice-in accordance 
with our more gcneral observations in Section 4.3 above.) 

But if Monty had decided only to reveal a door that was not hiding the prize, 
the situation is dramatically different. Suppose indeed he had decided that 
whenever Mary chose the white door, he would proceed as follows: if the prize 
is behind the white door, open the black one; but otherwise. open the door that 
does not hide the prize. Now, P(W nR*) = 0, so the posterior probability of the 
prize lying behind the white door is reduced to 0. Mary must change her 
choice-if she knows what Monty is doing. For in such circumstances. she 
would not have received her particular piece of new information if her initial 

choice had been a winner. 
Mere calculations (like these) do not, of course, remove the psychological 

and Jogical barriers to understanding the Monty Hall problem, but those 
difficulties should have been significantly reduced by our earlier analysis. 
As puzzles like this have been much debated elsewhere. I will not attempt a 
broader analysis here. My main point is simply to suggest that theseparadoxes 
are very much the product of a false conceptualization of the conditioning 
process; and, conversely, to indicate how easy it is to derive (and defend) some 



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694 Keith Hutchiscm 

of the confusing answers by applying my suggested revision of the misleading 
Rule. 

Acknowledgements 
I would like to thank two anonymaus referees for suggesting several improve-
ments to my discussion, notably: the relaxation of the requirement of persis-
tence at the end of Section 4.2, and the incorporation of a discussion of the 
traditional information-puzzles in Section 5. 

Department 4 History and Philosoph}· of' Science 
University of Me/b(JUme 

Parkville, 3052, Australia 

k.hutchison@ hps.unimelb. edu.au 

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