Consider the mini-roulette game presented in the classroom: each of six players
bet $ 1, independently of one another, on one of the six numbers 1, 2, ..., 6. A die is rolled, and if
its face up shows the number on which a player placed the bet, the player gets back their $ 1 plus
$ 4 more.
(a) Find the probability that all players bet on the same number.
(b) Find the probability that no two players bet on the same number (i.e., all bet on different
numbers).
(c) Find the expected value and the variance of the gain for the house in scenarios (a) and
(b) abov