There are many ways to bet your money on the outcome the Nevada roulette. Inf

addition to the popular Red-and-Black bet, here are two more bets.

. Single Number: If you bet a dollar on a single number at Nevada roulette, and that number comes up, you get the $1 back together with winnings of $35. If any other number comes up, you lose the dollar. Gamblers say that a single number pays 35 to 1, and there are 1 chances in 38 to win. . Split If you bet a dollar on a split at Nevada roulette. (A split is two adjacent numbers, like 11 and 12). If either number comes up, you gets the dollar back, together with winnings of $17. If neither number comes up, you loses the dollar. So a split pays 17 to I, and there are 2 chances in 38 to win. A gambler thinks 7 is very likely to come up, so he puts a dollar at 7. He also think 11 and 12 look promising, so he bets another dollar on 11 and 12 as a split. Let X be his net profit from the bet at a single number 17, and Y be his net profit from the bet at the split 11 and 12. The probability distributions of X and Y are as follows. Outcome 7 Other 37 numbers Outcome | 11 or 12 Other 36 numbers Value of X 35 Value of Y 17 probability 1/38 37/38 probability 2/38 36/38 (a) Find the expected value and the variance of X. (b) Find the expected value and the variance of Y. (c) If the gambler put the two bets in one spinning, let T = X + Y be his total net profit from the two bets. The probability distribution of T' can be found as follows. the other Outcome 7 11 or 12 35 numbers value of X 35 value of Y -1 17 value of T = X + Y 34 16 -2 Probability 1/38 2/38 35/38 Find the expected value and the variance of T. (d) Is the variance of T equal to the sum of the variance of X and the variance of Y? Explain briefly