Odds bets at craps. Refer to the odds bets at craps in given Exercise. Suppose that whenever the shooter has an initial roll of 4, 5, 6, 8, 9, or 10, he or she takes the odds. Here are the probabilities for these initial rolls:

Draw a tree diagram with the first stage showing the point rolled and the second stage showing whether the point is again rolled before a 7 is rolled. Include a first-stage branch showing the outcome that a point is not established. In this case, the amount bet is zero and the distribution of the winnings is the special random variable that has P(X = 0) = 1. For the combined betting system where the player always makes a $10 odds bet when it is available, show that the game is fair.

Exercise
A fair bet at craps. Almost all bets made at gambling casinos favor the house. In other words, the difference between the amount bet and the mean of the distribution of the payoff is a positive number. An exception is "taking the odds" at the game of craps, a bet that a player can make under certain circumstances. The bet becomes available when a shooter throws a 4, 5, 6, 8, 9, or 10 on the initial roll. This number is called the "point."

When a point is rolled, we say that a point has been established. If a 4 is the point, an odds bet can be made that wins if a 4 is rolled before a 7 is rolled. The probability of winning this bet is 1/3 and the payoff for a $10 bet is $20. You keep the $10 you bet and you receive an additional $20. The same probability of winning and payoff apply for an odds bet on a 10.

For an initial roll of 5 or 9, the odds bet has a winning probability of 2/5 and the payoff for a $10 bet is $15. Similarly, when the initial roll is 6 or 8, the odds bet has a winning probability of 5/11 and the payoff for a $10 bet is $12. Find the mean of the payoff distribution for each of these bets. Then confirm that the bets are fair by showing that the difference between amount bet and the mean of the distribution of the payoff is zero.