First, I suggest you take a look at this
problem, as you may decide that this is very similar! But here's a little twist.
Three men, Alan, Bob, and Charlie, were in separate cells under sentence of death when the governor decided to pardon one of them. He wrote their names on three slips of paper, shook the slips in a hat, drew out one of them, and telephoned the warden, requesting that the name of the lucky man be kept secret for several days. Rumor of this reached Alan. When the warden made his morning rounds, Alan tried to persuade the warden to tell him who had been pardoned. The warden refused.
"Then tell me," said Alan, "the name of one of the others who will be executed. If Bob is to be pardoned, tell me Charlie. If Charlie is to be pardened, tell me Bob. And if I'm to be pardoned, flip a coin to decide whether to name Bob or Charlie."
"But if you see me flip the coin," replied the wary warden, "you'll know that you're the one pardoned. And if you see that I don't flip a coin, you'll know that it's either you or the person I don't name."
"Then don't tell me now," said Alan. "Tell me tomorrow morning."
The warden, who knew nothing about probability theory, thought it over that night and decided that if he followed the procedure suggested by Alan, it would give Alan no help whatever in estimating his survival chances. So next morning he told Alan that Bob was going to be executed.
After the warden left, Alan smiled to himself at the warden's stupidity. There were now only two equally probable elements in the "sample space" of the problem. Either Charlie would be pardoned or himself, so by all the laws of conditional probability, his chances of survival had gone up from 1/3 to 1/2.
The warden did not know that Alan could communicate with Charlie, in an adjacent cell, by tapping in code on a water pipe. This Alan proceeded to do, explaining to Charlie exactly what he had said to the warden and what the warden had said to him. Charlie was equally overjoyed with the news because he figured, by the same reasoning used by Alan, that his own survival chances has also risen to 1/2.
Did the two men reason correctly? If not, how should each have calculated his chances of being pardoned.
(In reply to Solution
Alan's probability of survival is 1/3
Charlie's probability of survival is 2/3
Stealing from Penny's post:
(1) Bob to be spared, and warden says "Charlie" - 1/3
(2) Charlie is to be spared, and the warden says "Bob" - 1/3
(3) Alan is to be spared, the warden flips a coin, Charlie "wins", and the warden says "Charlie" - (1/2)(1/3) = 1/6
(4) Alan is to be spared, Bob "wins" the coin flip, and the warden says "Bob" -(1/2)(1/3) = 1/6
Now, the Warden says "Bob" so we can eliminate cases 1 and 3. The only possible cases are 2 (Charlie is to be spared 1/3 chance) and 4 (Alan is to be spared 1/6 chance). Because some possibilities were eliminated, our posibilities don't add up to 1 anymore, only 1/2. So we need to scale them up to 1 to restore our sense of probability. :) So...
Chance Charlie to be spared is 1/3 * 2 = 2/3
Chance Alan to be spared is 1/6 * 2 = 1/3
Looking at it a different way, we can ask "What information did Alan gain from the Warden's response?" The answer is none. Alan knew that the Warden was going to say "Bob" or "Charlie" regardless of whether Alan was to be pardoned, so how did he benefit from the response "Bob?" Alan's chance remained at 1/3 after he heard "Bob." However, Bob's chance dropped to 0 when the Warden said he wasn't going to be pardoned. Bob's 1/3 chance before the response went to Charlie after the response.
Posted by Russ
on 2004-02-08 04:36:41