Monty Hall problem

A decision problem associated with the American television game show host Monty Hall. Contestants are shown three closed curtains. Behind one is a prize, behind the other two are lemons. They pick a curtain. Monty Hall (who knows where the prize is) then pulls one of the other curtains, revealing a lemon, and contestants are asked if they would like to switch to the remaining curtain, or stay with their original choice. There seems to be no particular reason to switch, yet in fact switching doubles the chances of winning: your chance if you stay with your original curtain is what it always was, namely 1/3; the remaining curtain has a probability of containing the prize of 2/3. The problem was the subject of a minor scandal when several distinguished statisticians failed to see how this could be true. In fact it is true because there is now a significant difference between the curtain originally chosen, and the other one on offer, namely that Monty Hall avoided the second. The logic is more easily seen with a greater number of curtains. If there were 100, and you picked one, then by the time Monty Hall has pulled open 98 with lemons behind them, the chance that the remaining one that he did not pick conceals the prize is 99%. See also three prisoners, paradox of.