Discrete Mathematics
# Conditional Probability

You are in a game show! There are 10 closed doors: 9 lead to nothing and one leads to an expensive sports car. You are allowed to pick a door and earn the sports car if it's behind the door you choose. You choose a door and the host tells you he was preauthorized to make your chances of winning better! You have two options:

Option 1: Get the right to open two doors instead of one, and win if the car is behind either of the ones you open.

Option 2: Have the host open 5 empty doors (none of them the one you had chosen), and then get the right to switch if you want

What should you do?

A mathematician is on a gameshow, and the host gives him a choice of three doors; behind one is a Ferrari, but the other two lead to empty rooms. If he chooses the correct door, the host will open an empty door and give him the chance to choose again. However, if he chooses an incorrect door, the host will open the other empty door and give him the opportunity to choose again with probability $p$ (otherwise, he will tell him that he has lost).

The mathematician picks a door and the host opens another and gives him a chance to switch. The mathematician, who always makes true statements and is aware of the host's strategy, tells the host that changing does not improve or decrease his probability of winning the Ferrari. What is $p$?