(20 points) Let's play a variant of the game show problem. Imagine you are a contestant on a game show. The devious game show host offers you your choice of 4 doors total, and behind 2 of them is a prize. You pick a door. Without opening your door, the host opens another door to reveal there is nothing behind it. You are then given the option to pick one of the two remaining doors, or to keep the door you have chosen.

a. (5 points) What is the probability you choose a door hiding a prize with your initial guess?

b. (5 points) What is the probability you DO NOT chose a door hiding a prize with your initial guess?

c. (5 points) If you decide to pick one of the remaining two doors, what is the probability that you win a prize? Hint: It's not 1/2.

d. (5 points) Given that you decide to pick one of the two remaining doors and you do not win a prize, what is the probability that your initial choice was correct? Hint: Bayes' Rule.