Problem 6 [10 points = 4 + 6] Bob has $12 in his possession and decided to spend them playing in a casino. Any single game results in his loss with the failure rate p = 0.75, so his success rate is q = 1 p = 0.25. Results of individual games are independent. Bob wants to have n = 12 games played, each requiring a $1 fee. The number of games lost by Bob is denoted as Y, hence (12 - Y) is the number of his victories. When Bob wins, he gets the amount of $3, and when he loses, his one dollar fee is lost. Bob's balance after n = 12 games is B 3 (12 Y) - Y 1. Evaluate the expected balance for Bob, E [B] = 2. Find variance of B, Var [B] = Solution