Anthony and Barbara play the following game. First,
Barbara selects a card from an ordinary set of 52 playing
cards. Then, Anthony guesses which card Barbara
selected. If he guesses correctly, Barbara pays him 100
euros. If he guesses incorrectly, Anthony pays Barbara 4
euros. To make the game a bit more interesting, Anthony
is allowed to ask a yes/no question before he guesses,
and Barbara has to answer his question truthfully. Which
question is better: “Do you have a red card?” or “Do
you have the Queen of hearts?”?
The problem can be generalized a little bit more. While guessing one from a group of m has 1/m probability of success, isolating n members and first asking if it's one of them doubles your chances. The probability becomes:
(Yes, its one of n and then guessed) + (No, it's not and then guessed)
= (n/m) (1/n) + (m-n)/m 1/(m-n)
= 2/m
Here 2/52 = 1/26, so Ant is 25 times more likely to fail (25/26) then succeed, and the reward terms are set up 25:-1 to yield 0 mean. But that part is incidental. Since any question will always double the odds of success, any sample size and guessed sample size and any reward terms will yield an identical mean profit independently of the question.
a bit more generally (spoiler)
