Consider the following game:

First, a number N is drawn from the uniform distribution on the set {1, 2, 3, 4}. Then, a fair coin is flipped N times. If you flip at most one heads, you win.

• a) What is the probability that you will win?
• b) Let W be the indicator random variable of whether you win. Find the function g() that minimizes the expectation: E[ (W g(N) )² ]. (It is ok to specify your function g() by just defining the values of g(1), g(2), g(3), and g(4).)
• c) Find the function g() that minimizes the expectation E[ | W g(N) | ]. (That is, the function that minimizes the expectation of the absolute value of W g(N).)