Condition on the observation protocol, not just its words

Try

A box contains four distinct tokens A, B, C and D. Tokens A and B are marked. Choose an unordered two-token hand uniformly, so all six hands have probability 1/6. You hear, “There is a marked token.” Does that sentence tell you enough to calculate the chance that the hand contains both marked tokens?

Not until you know how the observation was produced. If someone inspected the hand and reported only whether it contained a marked token, five hands qualify and only one is AB. The answer is 1/5. But if one member of the hand was selected uniformly and that selected token was marked, AB has two chances to generate the observation while every mixed hand has only one. The answer will be 1/3.

These are two different experiments with the same initial distribution of hands. The ambiguity lies in the observation policy, not in the rules of conditional probability.

Predict

Before reading the table, list all six hands. Then, beside each, list the two equally likely selected tokens. The experiment “choose a hand, then choose one member uniformly” has twelve equally likely outcomes, each with probability (1/6)(1/2)=1/12. A hand and a selected member together form one outcome.

Containment information discards CD and leaves five equally likely hands; AB has conditional probability 1/5. A marked random reveal retains six of the twelve joint outcomes. Two have hand AB, so AB has conditional probability 2/6=1/3. Distinct marked labels A and B remain distinct outcomes even though the observed message conceals which label was selected.

Predict what happens if a reporter deliberately selects a marked token whenever possible, but reports only “marked.” That policy reports marked for every hand except CD with probability one, so it gives the containment posterior 1/5. It is not uniform selection from the hand.

Experiment

Use the table as a paper experiment. Circle every row consistent with containment. Separately circle every marked reveal cell. Count the AB outcomes in each selection and state the corresponding denominator aloud. Changing the denominator from five to six is justified by changing the elementary outcomes, not by making the hand larger.

Now change the information again: the uniformly selected token is specifically A. Which joint outcomes survive? They are (AB,A), (AC,A) and (AD,A), equally likely. The other hand member is uniform among B,C,D, so the chance it is marked is 1/3.

Changed-case calculation

Take five distinct tokens, two marked, and a three-token hand. There are ten hands. Exactly one has no marked token, six have one, and three have two. After containment information, the chance of two marked tokens is 3/9=1/3. After a uniform marked reveal, count marked cells across all hands: six from single-marked hands and six from double-marked hands. The posterior chance of two marked tokens is 6/12=1/2. These are fresh counts; the earlier 1/5 and 1/3 cannot be reused merely because there are still two marked tokens.

The calculations model a uniform hand and the explicitly specified policy. If a person chooses which token to show based on hidden information, ask for that rule before using either denominator.

Understand

Let N be the positive number of distinct tokens, K the number marked, and h the hand size, with integers 0≤K≤N and 1≤h≤N. Let J be the number of marked tokens in a uniformly chosen h-subset. Write C(n,r) for the number of r-subsets of n objects, using zero for impossible choices and C(0,0)=1.

To get J=j, choose j of the K marked tokens and h−j of the N−K unmarked tokens. These choices specify exactly one hand, and every such hand arises once. Therefore

P(J=j)=C(K,j)C(N−K,h−j)/C(N,h).

The feasible integers j run from max(0,h−(N−K)) through min(h,K). Outside that range the probability is zero. This is the hypergeometric law. It comes from sampling without replacement; the indicators of membership of distinct tokens generally are not independent.

Nevertheless expectation is straightforward by linearity. Each particular marked token is included in C(N−1,h−1) hands out of C(N,h), hence with probability h/N. Summing the K inclusion indicators gives E[J]=hK/N. No independence assumption is needed for that sum of expectations.

Let R be the event that a uniformly selected member of the hand is marked. Conditional on J=j, exactly j of the h equally likely choices are marked, so P(R given J=j)=j/h. Thus the joint probability of J=j and R is (j/h)P(J=j), while P(R)=E[J]/h=K/N. For K>0, division yields

P(J=j given R)=j P(J=j)/E[J].

This is a size-weighted posterior: hands with more marked members have more ways to generate the observation. By contrast, containment conditions on J≥1 and divides each positive-j probability by P(J≥1), without multiplying by j. When K=0 both observations have probability zero, so these posteriors are undefined, not distributions assigning zero to every count.

The same result has a direct counting explanation. Every joint hand/reveal outcome has probability 1 divided by h·C(N,h). A hand with j marked members contributes j favorable outcomes. The total number of favorable outcomes is K·C(N−1,h−1), by first choosing the marked token revealed and then its companions. This equals C(N,h)E[J]. Counting by hands or by revealed tokens gives the same normalizing factor.

A specified revealed token. Suppose the uniformly revealed token is one particular marked token A. Every hand containing A contributes exactly one such joint outcome, all with equal probability. Removing A leaves a uniform (h−1)-subset of the other N−1 tokens. Exactly N−K of those remaining tokens are unmarked. Consequently the chance of at least one additional marked token is

1−C(N−K,h−1)/C(N−1,h−1).

The denominator is positive for the stated parameter range. The observation that this particular token was selected has probability 1/N. If h=1, the ratio is one and there is no additional token. If the numerator requests too many unmarked tokens, it is zero and an additional marked token is certain. This proof uses uniform revealing; it does not authorize the same conclusion for an unspecified deliberate token-selection policy.

Apply

Suppose an inspection device has report likelihood wⱼ when the hand contains j marked tokens. The conditional law after its report is proportional to wⱼP(J=j), with normalizer equal to the sum of those products. A uniform reveal uses wⱼ=j/h. A deliberate “show marked if available” report uses w₀=0 and wⱼ=1 for positive j. Other policies produce other weights. The positive normalizer is part of the calculation, not a technical afterthought.

Test the boundaries. If every token is marked, J=h with certainty and both observations occur surely. If h=1 and K>0, either marked observation forces J=1, so the posteriors agree. If h=N and K>0, J=K is fixed: containment occurs surely, while a marked random reveal occurs with probability K/N. Their posterior laws of J agree even though their report probabilities differ. Agreement in a special case does not make the protocols identical.

A policy question before calculation

A reporter says “I found a marked token” after examining a hand. Before assigning a posterior, ask whether every hand is sampled uniformly, whether the reporter always announces a discovered mark, whether it sometimes withholds that report, and whether a selected token was chosen uniformly or deliberately. These determine the likelihood of the observation for each possible hand. The wording of the message alone does not determine those likelihoods.

Continue

In the practice, identify elementary outcomes before counting favorable cases. State what the observer does, which part of the observation is disclosed, and why the conditioning event has positive probability. Keep a hand’s membership separate from the act of selecting one member to reveal.

This lesson addresses containment, random revealing and changed reporting policies. It does not establish general results about joint independence or adaptive multi-stage stopping strategies. Those require their own sample spaces and proofs rather than treating one observation formula as a universal rule.