Problem:
Alice, Bob and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six.
Answer Choices:
A. 31​
B. 92​
C. 185​
D. 9125​
E. 9136​
Solution:
The probability that Carol tosses the first 6 on her initial throw is (65​)261​. The probability she tosses the first 6 on her second throw is (65​)561​. In general, the probability that the first 6 is tossed on Carol's nth throw is (65​)3n−161​. Therefore, the probability that Carol will be the first one to toss a 6 is
(65​)261​+(65​)561​+…+(65​)3n−161​+…
which is the sum of a geometric series with first term a=(65​)261​ and common ratio r=(65​)3. The sum is 1−ra​=9125​.