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.