Suppose that I roll a fair die (meaning that each number from 1 to 6 is equally likely) until I get a 6. What is the probability (in terms of k) that the first 6 appears the k^th time I roll the die? Suppose Alice and Bob play a game where the alternate rolling the die: First Alice rolls, then Bob, then Alice again, and so on. The first person to roll a 6 wins. What is the probability that Alice wins this game? Solution.

1. Suppose that I roll a fair die (meaning that each number from 1 to 6 is equally likely) until I get
a 6. What is the probability (in terms of k) that the first 6 appears the k^th time I roll the die?
Suppose Alice and Bob play a game where the alternate rolling the die: First Alice rolls, then
Bob, then Alice again, and so on. The first person to roll a 6 wins. What is the probability that
Alice wins this game?
Solution