# The Monty Hall Problem

In 1990 the following letter was sent into the ‘Ask Marilyn Column’ by Craig F. Whitaker from Columbia, Maryland:

“Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?”

Marilyn’s answer started out like this…

“Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance…”

Between 1986 and 1989 Marilyn vos Savant had been in the Guinness Book of Records with the highest IQ in the world, so you would think that people might respect what she had to say.  In fact – Marilyn’s answer generated a storm of 10,000 letters. (When things went viral back then you actually ended up with 25kg of paper coming though your door.)  Worse than that, she estimated that 1000 of the letters were from people with a PhD, and some were from respected university professors. Most were completely adamant that she was wrong and many arrogantly ridiculed those who agreed with her.

In fact this problem had been circulating long before Craig F. Whitaker sent it in to ‘Ask Marilyn’.  It had appeared in the 1970s where it was named after game show host Monty Hall who ran a similar game on his show ‘Let’s Make a Deal’.  Even before that an equivalent problem was talked about in the 19th Century.  So why did the Monty Hall Problem upset so many people and why is Marilyn actually right after all?

The Monty Hall Problem seems to challenge our common sense – if you have already picked a door then why does it matter if you switch? Some people pointed out that Marilyn hadn’t fully stated the exact constraints of the Monty Hall Problem – for example the fact that the game show host will always offer the chance to switch and that they will always open a door containing a goat and not the car.    The readers who were arguing with Marilyn weren’t debating the finer points of how she had stated the problem – they just didn’t believe that there was an advantage to switching!

Right now you might be on the side of the 1000 PhD qualified people who disagreed with Marilyn - if you still don’t feel convinced about the advantages of switching then consider the following:

One third of the time: You will have picked the car (you don’t know this) but you should stick with your door.

Two thirds of the time: You will have picked a goat (again you don’t know this), and the host will then open up the door with the other goat. So you should switch and you will get the car.

Therefore one third of the time you should switch – if you play the game 99 times and don’t switch then you will probably lose around 66 times, whereas if you switch every time you will probably win around 66 times.

If you are feeling a bit red faced about disagreeing with Marilyn, then don’t worry – the feeling that you shouldn’t switch runs deep - in one research study run by psychologists it was found that only 13% of people naturally wanted to switch.

There are lots of possibilities for using Monty Hall in the classroom, particularly as an investigation. Ask your pupils to play the game and log their results.  How many games are played before the results start to back up Marilyn?  Playing Monty Hall can open up conversations about conditional probability and the confusion it can create, and pupils could even end by creating their own simple game show based on Monty Hall. Consider how changing the rules affects the best strategy – the possibilities are endless.

#### Extra resources:

Monty Hall Simulation https://www.mathwarehouse.com/monty-hall-simulation-online/