Remember the game show “Let’s Make a Deal”? This problem is named after the host of that show. During the show, a contestant would be presented with three doors to choose from. One of the doors had a valuable prize behind it like a Caribbean vacation or a car. The other two doors had some less desirable prizes like a month’s supply of Meow Mix or the collective works of Menudo. The host, Monty Hall, knew what was behind each door.

The contestant would be asked to pick one of the doors. Monty would then opened one of the doors that the contestant did not pick to reveal one of the crummy prizes. He'd then asked the contestant if he/she would like to switch their choice to the other unopened door.

Is it advantageous for the contestant to switch?

On the face of it, the obvious answer seems to be that it wouldn’t make any difference because the odds are 50/50. You have two doors and one of them has the good prize. Interestingly enough, that is wrong. The contestant is twice as likely to win if they switch! The disparity arises because Monty knows what is behind the doors and has actively interfered with blind chance. It all stems from the contestant’s first choice.

In the “lucky first choice” scenario, the contestant picks the door with the valuable prize behind it. Monty opens either of the other two doors because either one has crap behind it. In this scenario, if the contestant switches, they lose. Because there is one good door and two crap doors, this scenario has a 1/3 probability.

In the “unlucky first choice” scenario, the contestant picks one of the two doors with crap behind it. Monty then opens the other door with crap behind it, deliberately leaving the valuable door shut. In this scenario, the contestant will win if they switch. Because there is one good door and two crap doors, this scenario has a 2/3 probability.

So, when Monty asks contestants if they want to switch doors, he is actually asking them if they want to switch their odds of loosing with their odds of winning, and the answer should be, “Hell yeah!” To reinforce this point, a statistical analysis was done on the show. The results showed that the contents that switched did, in fact, win about twice as often than the ones that didn’t.