Exercise C.2.9

$\star$ You are a contestant in a game show in which a prize is hidden behind one of three curtains. You will win the prize if you select the correct curtain. After you have picked one curtain but before the curtain is lifted, the emcee lifts one of the other curtains, knowing that it will reveal an empty stage, and asks if you would like to switch from your current selection to the remaining curtain. How would your chances change if you switch? (This question is the celebrated Monty Hall problem, named after a game-show host who often presented contestants with just this dilemma.)

This is very popular.

If you are always given this choice and take it:

• If you picked the right curtain (probability $1/3$), you will switch to an empty stage
• If you picked a wrong curtain (probability $2/3$), you will switch to the prize (that's the only option, since the other empty stage is shown)

Effectively, you interchange success and failure. That way, the probability to win is $2/3$ and you should always take it (if the choice is given to you in an unbiased fashion).