They suppose the following question. Please show all steps. I really want to look at my logic when compared to someone else.

Question 4 . A race-car driver wants to model the chance of winning a race. If she gets 8 or more hours of sleep, then there is a guaranteed win. Let ? = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} represent the amount of hours of sleep the race-car driver received the night before. We will define our random variable X to be a zero-one random variable. That is, it takes the value 1 if the race-car driver wins and O if the race-car driver looses. Unfortunately, the outcomes on S are not equally likely. They have the following probabilities: W 0 1 2 3 4 5 6 7 8 9 10 11 12 P({w} ) 1 1 2 2 4 8 20 20 24 8 8 1 1 100 100 100 100 100 100 100 100 100 100 100 100 100 1. What is the probability that the race-car driver wins the race? Calculate P(X = 1). Similarly, calculate P(X = 0) the race-car driver looses the race she set out to win, by not getting enough sleep the night before. 2. | What is E[X]? how does that compare with the probability from part 1? 3. | Define a new random variable Y = 1 - X. That is, Y = 1 when X = 0 and Y = 0 when X = 1. Find the probability mass function of Y and calculate its expectation (Hint: the PMF of Y can be easily derived from the PMF of X)