Question

2. (From the Fall 2017 takehome.) In American roulette, a roulette wheel with 38 possible outcomeste numbers 1 to 36, "0", and "00" is spun. As the wheel spins, players makes bets on which number the ball w land on. Eventually the wheel stops, and a ball lands randomly on one of the 38 numbers, with an equal probability for each one. A statistics professor's father bets that the wheel wil land on one of the numbers from 1 to 12. The happens with probability 12/38. If this happens, the player wins $2. Otherwise, the player loses $1 (a) Suppose the roulette player makes this bet once. Let X be a random variable represent- ing the number of times the player wins. What are the expected value and standard deviation of X? (b) Suppose the roulette player makes this bet 500 times. Let Y be a random variable representing the number of times the player wins. What are the expected value and standard deviation of Y? (c) For the player to make a profit over 500 bets, the wheel must land on the numbers from 1 to 12 on at least 167 spins. What is the probability the player makes a profit?

Answer 1