The game Chuck-A-Luck is played as follows: A player bets $M on an integer chosen from 1-6 and then rolls three dice. If the number appears on exactly on die, then the player wins the amount bet. The player wins twice the amount bet if the umber appears on two dice and three times the amount bet if the number appears on all three dice. If the number does not appear at all, then the player loses the $M. What is the probability of winning at each level? At any level? For every n such bets of a fixed amount $M, what would be the House's average net gain? (In other words, find u such that u = average gain by the house).

Answer 1

Whatever number is chosen, there is a 1/6 chance of that showing up on each die.

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

P(win one) = 1/6 * 5/6 * 5/6 = ?

P(win none) = 5/6 * 5/6 * 5/6 = ?

I'll let you do the rest.