Problem:
Alex, Mel, and Chelsea play a game that has 6 rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is , and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
If Alex wins 3 rounds, Mel wins 2 rounds, and Chelsea wins 1 round, then the game's outcomes will be a permutation of AAAMMC, where the letter represents the initial of the winner of the round. There are
such permutations.
Because each round has only one winner, it follows that . Also and so and .
The probability that Alex wins 3 rounds, Mel wins 2 rounds, and Chelsea wins 1 round is therefore
The problems on this page are the property of the MAA's American Mathematics Competitions