# Monty Hall Problem

## 27/2/19

The Monty Hall question arose from a US TV quiz show in the 1950's, hosted by Monty Hall. In the quiz, the contestants were given a choice of three boxes and were told to choose one. One box had a prize, the other two were empty. Once the chosen box was identified (box A), Hall (who was aware which box had the prize in) would open one of the other two boxes (box B) that he knew did not contain the prize. The contestants where then asked if they wanted to stick with their original choice, or change to the third unopened box (Box C). The question now is, using probability, what should the contestant do? Stick or change?

The intuitive answer is that it doesn't matter. you still retain the same 1/3 chance of being correct from the original choice. However, mathematically the contestant should always swap and go with box C.

If you do not switch, you retain the original 1/3 chance of being correct. However, because hall has revealed Box B does not have the prize, you now have more information and by choosing box C your odds are 2/3 to being correct - Box A by itself is 1/3, Box B & C together are 2/3 odds of being right. By removing Box B for you, Box C retains those same odds.