mathematics

Three prisoners problem

Three prisoners problem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Three prisoners problem rather than just read about it. In short: The three prisoners problem appeared in Martin Gardner's "Mathematical Games" column in Scientific American in 1959. It is mathematically equivalent to the Monty Hall problem with the car and goat replaced respectively with freedom and execution.

Key takeaways

Three prisoners problem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Three prisoners problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Three prisoners problem from memory before moving on to harder problems.

Reference excerpt

The three prisoners problem appeared in Martin Gardner's "Mathematical Games" column in Scientific American in 1959. It is mathematically equivalent to the Monty Hall problem with the car and goat replaced respectively with freedom and execution.

Problem Three prisoners, A, B, and C, are in separate cells and sentenced to death. The governor has selected one of them at random to be pardoned. The warden knows which one is pardoned, but is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the two who are going to be executed. "If B is to be pardoned, give me C's name. If C is to be pardoned, give me B's name. And if I'm to be pardoned, secretly flip a coin to decide whether to give me name B or C." The warden gives him B's name. Prisoner A is pleased because he believes that his probability of surviving has gone up from ⁠1/3⁠ to ⁠1/2⁠, as it is now between him and C. Prisoner A secretly tells C the news, who reasons that A's chance of being pardoned is unchanged at ⁠1/3⁠, but he is pleased because his own chance has gone up to ⁠2/3⁠. Which prisoner is correct?

Solution The answer is that prisoner A did not gain any information about his own fate, since he already knew that the warden would give him the name of someone else. Prisoner A, prior to hearing from the warden, estimates his chances of being pardoned as ⁠1/3⁠, the same as both B and C. As the warden says B will be executed, it is either because C will be pardoned (⁠1/3⁠ chance), or A will be pardoned (⁠1/3⁠ chance) and the coin to decide whether to name B or C the warden flipped came up B (⁠1/2⁠ chance; for an overall ⁠1/2⁠ × ⁠1/3⁠ = ⁠1/6⁠ chance B was named because A will be pardoned). Hence, after hearing that B will be executed, the estimate of A's chance of being pardoned is half that of C. This means his chances of being pardoned, now knowing B is not, again are ⁠1/3⁠, but C has a ⁠2/3⁠ chance of being pardoned.

Table The explanation above may be summarised in the following table. As the warden is asked by A, he can only answer B or C to be executed (or "not pardoned").

As the warden has answered that B will not be pardoned, the solution comes from the second column "not B". It appears that the odds for A vs. C to be pardoned are 1:2.

Mathematical formulation Call A {\displaystyle A} , B {\displaystyle B} and C {\displaystyle C} the events that the corresponding prisoner will be pardoned, and b {\displaystyle b} the event that the warden tells A that prisoner B is to be executed, then, using Bayes' theorem, the posterior probability of A being pardoned, is:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Three prisoners problem

Start with the simplest possible case. Write down what Three prisoners problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Three prisoners problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Three prisoners problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Three prisoners problem

In research: Three prisoners problem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Three prisoners problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Three prisoners problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Decision-making paradoxes, Probability problems, Probability theory paradoxes, so understanding it makes those chapters shorter.
In everyday life: Look for Three prisoners problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Three prisoners problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Three prisoners problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Three prisoners problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Three prisoners problem in simple terms?

Why does Three prisoners problem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Three prisoners problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Three prisoners problem.