On Let’s Make A Deal (on which Monty Hall was the original host), contestants were shown three doors and told that behind one of the doors was a brand new car. If they could correctly guess which door hid the car, they could keep it. The guessing happened in two phases.
First the contestant would choose a door that they believed had the car behind it. Then one of Monty Hall’s assistants would walk over to the other two doors (the ones that had not been chosen) and proceed to open one of them, always revealing a bad prize (often a donkey!). Then the contestant, who now had this new information, was allowed to stick with their original answer or they could switch to the remaining closed door. That would be the final allowed guess.
They’d open the final chosen door and their winnings (hopefully a car!) would be displayed.
In 1990, well into the show, advice columnist Ask Marilyn from Parade Magazine was asked whether there was a particular advantage to staying or switching. Much to many folks’ surprise at the time, she answered that the probability of winning was greater if you switched. Thousands of people wrote in to tell her that she was wrong, but most (not all!) were eventually convinced.
My best summarization would be that, initially, each door has 1/3 probability of being correct. Therefore, the door that the contestant first chooses has a 1/3 chance of being correct, and the sum of other 2 choices must be 2/3. When the assistant reveals the donkey behind one of the remaining doors, it does not change these facts. The original choice still has a 1/3 chance of being correct, and the sum of the other doors still has a 2/3 chance of being correct. However, now that the contestant knows that one of the non-chosen doors is a bad door, the 2/3 chance must lie solely in the unchosen, unopened door.
The conclusion is that you have a 2/3 chance of guessing correctly if you switch, and you have a 1/3 chance of being correct if you stay. This has since been mostly proven as well as observed in repeated computer simulated trials. It’s a bad interview question, but remains a popular Probability 101 problem and a decent anecdote.