There are ten boxes, one of which contains a $100 bill, and the other nine boxes are empty. Now let you choose one of the boxes, open it, if there is a banknote inside, this banknote belongs to you, when you choose the box, the host opens 8 empty boxes without banknotes, leaving the last box, at this time, do you want to exchange the box in your hand with the one that is not opened?
For such a game, intuition tells us that if you choose one out of ten, the probability that one of the boxes has banknotes is 1 10, and if you open 8 boxes and become one of the two, then no matter which one you choose, the probability of getting the banknotes is 1 2, so it doesn't matter if you change the box or not. This explanation seems to make sense and conforms to our thinking, but it is not the case.
Suppose we take out two bags, first select one box at random and put it in the bag, and then put the other 9 boxes into the other bag, then you only have one box in the bag, would you like to exchange the bags?Obviously, a lot of people will choose to swap bags. In the first case, after opening 8 boxes, it is 1 to 1, which makes people mistakenly believe that it is the probability of 1 2, and the human analysis and judgment is wrong. The second case is 1 to 9, obviously the probability of having banknotes in bags of 9 boxes is higher, and choosing to exchange will change the probability of getting banknotes from 1 10 to 9 10.