2. In a roulette game, the ball is equally likely to fall in any one of the 38 slots. You will bet on 4 numbers. A roulette bet on 4 numbers has a payoff of 8:1. You have doubled your bet from $5 to $10.

(a) Show the probability distribution of the $10 bet as a table;

(b) Graphically, show how the probability distribution of the doubled bet is related to the original bet of $5;

(c) Compute the expected value of the doubled bet from (a);

(d) Check that the expected value of the doubled bet is in accordance with the expected value of a discrete random variable theory, involving the expected value of the $5 bet;

(e) Compute the variance of the $10 bet from the probability distribution in (a);

(f) Check that the results of (e) are properly related to the variance of the $5 bet.