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 0.5 and the payoff for a $10 bet is $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 2/3 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 5/6 and the payoff for a $10 bet is $12. Find the mean of the payoff distribution for each of these bets. Then con?rm that the bets are fair by showing that the difference between amount bet and the mean of the distribution of the payoff is zero.