Problem:
Alice, Bob, and Carol play a game in which each of them chooses a real number between and . The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between and , and Bob announces that he will choose his number uniformly at random from all the numbers between and . Armed with this information, what number should Carol choose to maximize her chance of winning?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because Alice and Bob are choosing their numbers uniformly at random, the cases in which two or three of the chosen numbers are equal have probability 0 and can be ignored. Suppose Carol chooses the number . She will win if her number is greater than Alice's number and less than Bob's, and she will win if her number is less than Alice's number and greater than Bob's. There are three cases.
If , then Carol's number is automatically less than Bob's, so her chance of winning is the probability that Alice's number is less than , which is just . The best that Carol can do in this case is to choose , in which case her chance of winning is .
If , then Carol's number is automatically greater than Bob's, so her chance of winning is the probability that Alice's number is greater than , which is just . The best that Carol can do in this case is to choose , in which case her chance of winning is .
Finally suppose that . The probability that Carol's number is less than Bob's is
so the probability that her number is greater than Alice's and less than Bob's is . Similarly, the probability that her number is less than Alice's and greater than Bob's is . Carol's probability of winning in this case is therefore
The value of a quadratic polynomial with a negative coefficient on its quadratic term is maximized at , where is the coefficient on its quadratic term and is the coefficient on its linear term; here that is when , which is indeed between and . Her probability of winning is then
Because the probability of winning in the third case exceeds the probabilities obtained in the first two cases, Carol should choose .
The problems on this page are the property of the MAA's American Mathematics Competitions