Problem 44 - A First Course in Probability

Question

Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner \(A\) asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already knows that at least one of the two will go free. The jailer refuses to answer the question, pointing out that if \(A\) knew which of his fellow prisoners were to be set free, then his own probability of being executed would rise from \(\frac{1}{3}\) to \(\frac{1}{2}\) because he would then be one of two prisoners. What do you think of the jailer's reasoning?

Step-by-Step Solution

Answer

The jailer's reasoning is incorrect because the conditional probability of A being executed, given that B is set free, is still \(\frac{1}{3}\) and not \(\frac{1}{2}\) as the jailer claims. The probability for A being executed doesn't change if the jailer reveals the information about one of the fellow prisoners being set free.

Step 1: Initial Probabilities

Initially, since the execution decision is random, the probabilities for each prisoner being executed are equal; that is, for each prisoner, the probability of being executed is \(\frac{1}{3}\). Formally, we can write: - \(P(A) = P(B) = P(C) = \frac{1}{3}\) Now, let's consider what would happen if the jailer tells prisoner A which of his fellow prisoners (B or C) will be set free.

Step 2: Conditional Probabilities

Suppose the jailer tells A that B will be set free (without the loss of generality, as the same reasoning would apply if the jailer tells A that C will be set free). The situation for prisoner A then becomes a new conditional probability problem. We have to calculate the probability of A being executed, given that B is set free. We can write this as: - \(P(A|B')\) (Here, \(B'\) denotes the event that B is set free.) We will use the Bayes' theorem to calculate this probability: \[ P(A|B') = \frac{P(B'|A)P(A)}{P(B')} \] To find \(P(B'|A)\), we know that if A is executed, then either B or C will be set free. So we have: - \(P(B'|A) = \frac{1}{2}\) Next, we need to find the probability P(B'), that is B will be set free regardless of who will be executed. To find P(B'), we consider all possible outcomes where B is set free: 1. A is executed, and B, C is set free: \(P(A)P(B'|A) = \frac{1}{3}\cdot\frac{1}{2}\) 2. C is executed, and A, B is set free: \(P(C)P(B'|C) = \frac{1}{3}\cdot1\) So the unconditional probability of B being set free is: - \(P(B') = P(A)P(B'|A) + P(C)P(B'|C) = \frac{1}{3}\cdot\frac{1}{2} + \frac{1}{3}\cdot1 = \frac{1}{2}\) Plugging these values into the Bayes' theorem, we get: - \(P(A|B') = \frac{\frac{1}{2}\cdot\frac{1}{3}}{\frac{1}{2}} = \frac{1}{3}\) The probability of A being executed, given that B is set free, is still \(\frac{1}{3}\). Therefore, the jailer's reasoning is incorrect, and the probability for A being executed doesn't change to \(\frac{1}{2}\) if the jailer reveals the information about one of the fellow prisoners being set free.

Key Concepts

Bayes' Theorem

Bayes' Theorem is a powerful statistical tool used to update the probability of a hypothesis based on new evidence. It is especially useful in cases where we want to find conditional probabilities—these are the chances of an event occurring given that another event has happened.
In the context of the "Prisoner Problem", Bayes' Theorem helps us to determine the probability of Prisoner A being executed given new evidence, namely if either Prisoner B or C is set free. Specifically, if we learn that B is set free, we calculate the probability of A being executed using the formula:
\[ P(A|B') = \frac{P(B'|A)P(A)}{P(B')} \]
This equation breaks down as follows:

• \(P(A|B')\) is the probability of A being executed given that B is released.
• \(P(B'|A)\) is the probability of B being released if A is executed.
• \(P(A)\) is the baseline probability of A being executed, initially \(\frac{1}{3}\).
• \(P(B')\) is the probability that B is released based on all possible scenarios.

By applying Bayes' Theorem, we can systematically update our beliefs regarding the fate of Prisoner A based on disclosed information about the other prisoners.

Probability Calculation

Probability calculation is fundamental to understanding the likelihood of events occurring. When dealing with problems that involve randomness, such as the execution of one prisoner out of three, precise probability calculations help clarify the potential outcomes.
Initially, each prisoner has a \(\frac{1}{3}\) chance of being executed since the decision is random. To complicate this, when new information is introduced, like the jailer revealing who is not executed, it modifies our calculations by introducing conditional probabilities.
For example, after we calculate \(P(B')\)—the likelihood that B is set free—we recognize that for each possible execution scenario, probabilities must be weighted by their respective initial probabilities \(P(A)\) and scenarios, like \(P(B'|A) = \frac{1}{2}\).

• If A is executed, either B or C is set free, leading to \(P(B'|A) = \frac{1}{2}\).
• If C is executed, B is always set free, giving \(P(B'|C) = 1\).

These calculations reinforce that understanding both initial and new conditional probabilities are essential for accurate determination of outcomes.

Prisoner Problem

The Prisoner Problem is a classic thought experiment that demonstrate the use of conditional probability. The issue lies in assessing the probability of one event occurring when given certain new conditions or constraints.
Initially, the problem is simply: since one of the three prisoners—A, B, or C—is to be executed at random, each has a \(\frac{1}{3}\) probability of being chosen. However, the twist comes when we try to reason out what happens to these probabilities if one prisoner finds out who will not be executed.
The jailer's statement to Prisoner A includes flawed logic: A assumes knowing that either B or C will be freed changes his probability of execution from \(\frac{1}{3}\) to \(\frac{1}{2}\).
However, through both the correct application of probability calculations and Bayes' Theorem, we find that despite additional information being released, the conditional probability of A still remains \(\frac{1}{3}\).
This reveals how intuitions about probability and information can often be misleading without proper consideration of the probabilistic framework that accounts for all possible outcomes.