If you feel comfortable that the answer to the Monty Hall problem is "switch", you might want to test your understanding to be sure you reach the correct answer here as well. As Wikipedia states, Three Prisoners "is mathematically equivalent to the Monty Hall problem with car and goat replaced with freedom and execution respectively". But mathematically equivalent may not make it intuitively equivalent, and the answer may feel like it contradicts the correct Monty Hall answer.
Wikipedia points out that it's originally from a 1950's Martin Gardner column, but I came across it in the textbook "Stastistical Inference" by Casella and Berger. Here's C&B's phrasing:
Three prisoners, A, B, and C, are on death row. The governor
decides to pardon one of the three and chooses at random the
prisoner to pardon. He informs the warden of his choice but
requests that the name be kept secret for a few days. The
next day, A tries to get the warden to tell him who had been
pardoned. The warden refuses. A then asks which of B or C
will be executed. The warden thinks for a while, then tells A
that B is to be executed.
Warden’s reasoning: Each prisoner has a 1 in 3
chance of being pardoned. Clearly, either B or C must be
executed, so I have given A no information about whether A
will be pardoned.
A’s reasoning: Given that B will be executed, then either A
or C will be pardoned. My chance of being pardoned has
risen to 1 in 2.
Who is right?
In what I found to be a parody of textbook tropes, C&B begin their explanation "It should be clear that the warden's reasoning is correct..."
While it's true that the warden's reasoning is correct, leading off with "it should be clear" seems cruel. Here's a slight variation of the Monty Hall problem, one of the most famous "paradoxes" of popular statistics, and you are going to start with a paraphrase of "it should be obvious to the reader" without the slightest sense of irony, even though this variation produces an answer superficially incompatible with the better known problem? Ah, the strange humor of textbook authors! The remainder of their answer (which is solid) can be found here on Section 1.3 page 22: http://people.unica.it/musio/files/2008/10/Casella-Berger.pd...
If you feel comfortable that the answer to the Monty Hall problem is "switch", you might want to test your understanding to be sure you reach the correct answer here as well. As Wikipedia states, Three Prisoners "is mathematically equivalent to the Monty Hall problem with car and goat replaced with freedom and execution respectively". But mathematically equivalent may not make it intuitively equivalent, and the answer may feel like it contradicts the correct Monty Hall answer.
Wikipedia points out that it's originally from a 1950's Martin Gardner column, but I came across it in the textbook "Stastistical Inference" by Casella and Berger. Here's C&B's phrasing:
In what I found to be a parody of textbook tropes, C&B begin their explanation "It should be clear that the warden's reasoning is correct..."While it's true that the warden's reasoning is correct, leading off with "it should be clear" seems cruel. Here's a slight variation of the Monty Hall problem, one of the most famous "paradoxes" of popular statistics, and you are going to start with a paraphrase of "it should be obvious to the reader" without the slightest sense of irony, even though this variation produces an answer superficially incompatible with the better known problem? Ah, the strange humor of textbook authors! The remainder of their answer (which is solid) can be found here on Section 1.3 page 22: http://people.unica.it/musio/files/2008/10/Casella-Berger.pd...