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## Monty Hall Problem

Trying to find the best strategy to win in this game is a famous brain teaser, not least because so many people (including mathematicians) get it wrong.

The game is very simple: you are shown three closed doors. Behind one of them is a car, behind the two others, a cow. You first pick one door, but it does not open right away. The game host, Monty Hall, who knows behind which door the car is waiting, then teases you by opening one of the two doors that you had not picked to show you the cow sitting there. Then he may offer you a choice: staying with your original pick, or switching to the third remaining door. Should you switch or shouldn't you?

Well, it really all depends!

On the next page, you'll play this game in three different versions, with three different hosts.

Host 1 always offers you the possibility to switch after he has opened one of the doors you didn't pick which has a cow.

Host 2 offers that choice only some of the time; his decision to do so depends on the door you picked on the first go. You'll have to find out what his rule for deciding is.

Host 3 decides randomly, every time, whether he will behave like Host 1 or 2. That is, before each round, he flips a fair coin. If it comes up heads he'll behave like host 1. If it comes up tails, he'll behave like host 2.

Play the game a number of times with each host, and try to decide which strategy (switching or not) is the best. Pick the host before each turn; if you don't choose one, you will have the same host as in the previous turn. Next to each host, you'll see a tally totalling:

1) the number of times played
2) the number of times a switch was offered
3) the number of games won after switching
4) the number of games lost after switching
5) the number of games won after no switch happened (not offered, or offered but not taken)
6) the number of games lost after no switch happened (not offered, or offered but not taken)

These tallies will help you to do your problem set. WARNING: However, you need to keep track of one additional number mentally---keep track of the number of times that no switch was offered and you won.