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Conditional Probability

Conditional probability is the art of updating probabilities based on given information. What is the probability that the sidewalk is wet? And what if we know that it rained a few hours ago?

Monty Hall

You are in a game show! There are 3 closed doors: two lead to nothing and one leads to 300 dollars (there wasn't much funding for the game show that year). You get to choose a door and receive whatever is behind it. However, after you pick, the game show host (who knows where the 300 dollars are) has been instructed to open an empty door you didn't pick and sell you the chance to switch to the other unopened door (he wants to make money for next year's game show). If you want to maximize the expected value of your earnings, what is the highest price you should buy the chance to switch at?

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 $$p$$?

There are six doors lined up in a row. Two adjacent doors both have an expensive prize, and the other four have nothing. You pick the third door, but then the host (who you know will open the left-most door that is empty and you haven't chosen) opens the first door. If you then decide to switch to the fourth door, what is the probability that you got the prize?

You are on a game show, and must correctly identify the one door of three that has the grand prize. You pick a door, and then the host asks you to pick another door which he may or may not show you the contents of. You point to a door, and the host refuses to open it. You know that if the door you had pointed to had been the door with the prize, the host would have never opened it, and if it had been empty, he would have opened it half the time. The host then offers you the chance to switch doors (from your original choice, not the second door you pointed at) if you want, but you refuse and keep the first door you chose. What is the probability you will get the prize?

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