Brainteaser: The Monty Hall Problem

You are on a game show and presented with 3 doors.  Behind one is a car and behind the other 2 are goats.  You want to choose the door with a car behind it, as if you do so, you win the car.  You choose one door.  Then, the host opens one of the other doors, which reveals a goat behind it.  The host gives you a choice to either switch your door to the other one that’s still closed or keep your original choice.  Should you switch doors?

 


 

Answer:

If your strategy is to stick to your original choice, your probability of choosing the door with the car behind it is 1/3.  Let’s see what happens if you switch.  So you choose a door, the host reveals one of the other doors with a goat behind it, and asks if you want to switch.  What has happened up to this point?  There’s a 1/3 chance that you picked the door with the car behind it, which means that if you switch, you are switching to a door with a goat behind it.  There’s a 2/3 chance that you picked a door with a goat behind it, which means that if you switch, you are switching to a car behind it.  So if your strategy is to always switch, there’s a 1/3 chance you get a goat in the end (because you happened to choose a car on your first choice, which has a probability of 1/3) and a 2/3 chance you get a car in the end (because you happened to choose a goat on your first choice, which has a probability of 2/3).  So the best strategy is to switch.

The host revealing one of the doors gives you additional information.  Switching lets you use that information, assuming that it was unlikely that you got a car on your original choice.

Perhaps a more intuitive answer is if there are 100 doors.  One has a car behind it and 99 of them have goats behind them.  Choose one door, the hosts reveals another door with a goat behind it, and asks if you want to switch.  If you don’t switch, there’s a 1/100 chance that you chose the door with a car behind it.  But if you switch, assuming that you probably didn’t choose the right door on your first try (because 1/100 is small), now, you have a 1/98 chance of choosing the right door (because the host as revealed one door with a goat behind it and you’re giving up your original door).  Of course 1/98 is better than 1/100.  The exact probability of getting the right door with the switching strategy is 99/100 × 1/98 (probability that you chose the wrong door on the first try × probability of choosing the right door after accepting the offer to switch).  99/100 × 1/98 = 1/100 × (99/98) > 1/100 where 1/100 is the probability of getting the car with not switching, and so switching is better than not switching.

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