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 questions are equally good and the game is fair. In fact, any question that elicits information about Barbara's holding is equally good, which is a surprising result.
Analysis follows:
(a) Do you have a red card?
Prob of a Yes = 1/2, after which prob of guessing correctly = 1/26
Prob of a No = 1/2, after which prob of guessing correctly = 1/26
Expected chance of guessing correctly = (1/2)*(1/26) + (1/2)*1/26) = 1/26
(b) Do you have the Queen of hearts
Prob of a Yes = 1/52, after which prob of guessing correctly = 1
Prob of a No = 51/52, after which prob of guessing correctly = 1/51
Expected chance of guessing correctly = (1/52)*(1) + (51/52)*1/51) = 1/26
(c) A question which elicits info and has probability p of being yes:
Prob of a Yes = p, after which prob of guessing correctly = 1/52p
Prob of a No = (1-p), after which prob of guessing correctly = 1/(52*(1-p))
Expected chance of guessing correctly = (p)*(1/52p) +(1-p)/(52*(1-p)) = 1/52 + 1/52 = 1/26
EXPECTED WINNING WITH ANY QUESTION =
100*(1/26) + (-4)*(25/26) = 0
Very interesting! (spoiler)
