The stats puzzle that defeated mathematicians

Maths may be very important, but it rarely makes headline news. But the Monty Hall problem, a seemingly simple statistics puzzle, is one that proved hugely contentious, busting the brains of scientists since 1975. Even Paul Erdos, the renowned mathematician, fell victim to the mistaken assumptions that complicate this problem. When we look at the Monty Hall problem, we can see that it’s often incredibly hard to push our brains to ignore the obvious.

The problem, named after the game show Let’s Make a Deal and its host Monty Hall, goes something like this. Imagine you’re on a game show, and you must choose one of three doors – you will win whatever is behind that door. Behind one of the doors is a car, and behind the other two, there are goats. After you’ve chosen a door, the host opens one of the remaining two to reveal a goat. So, we’re now in a situation in which one door conceals the car, the other, the goat. The host gives you the opportunity to swap your door to the other remain closed one. The question is this – is it to your advantage to switch doors?

I’ll give you a few moments to think about the answer, and provide a little history while you do. Although variants of the problem had existed for a hundred years prior, it was a 1990 edition of Parade Magazine that helped it hit the headlines. Marilyn vos Savant, a child prodigy who became renowned as the world’s smartest woman, wrote a column in which she answered a variety of academic questions and logic puzzles. But ‘Ask Marilyn’ became the subject of huge controversy after vos Savant took on the Monty Hall problem – she was subjected to a massive letter-writing campaign, in which she was mocked as an idiot and belittled for her gender.

It’s a tricky thing to move your brain on from the initial incorrect assumption

What is the solution, then? Should you stick with your original door or switch to the other unopened door? The correct answer, and the one that caused vos Savant such trouble, is that you should switch – your original door has a 1/3 chance of winning, while the second one has a 2/3 chance. You’re far likelier to win the car if you swap your door, but understanding exactly why is a challenge. Think of the game as one of two stages. In the first, you must randomly choose a door – in this case, whichever door you choose, you have a 1/3 chance of your chosen door concealing the car. Thus, consequently, the remaining two doors have a 2/3 chance of containing the car – so far, so simple.

It’s the next stage that causes the trouble. The host opens one door, revealing a goat, and leaving the puzzle as a two-door game. The immediate and obvious assumption is that we’ve now begun a new game – one door conceals a car, one a goat, and so you’ve a fifty-fifty chance of winning with either door, whether you stick or swap. But this assumption is wrong – the host opening one of the doors doesn’t start a new game. The host revealing one of the goats only recontextualises the problem, and this is only the case because the host knows where the car is. The odds that your initial door conceals the car are still 1/3, and it’s still 2/3 likely that the car is behind one of the two other doors. But now you know that one of the other doors is a goat, the 2/3 odds transfer to the door you could switch to. So, it’s 1/3 you’ll win with the original door, 2/3 odds if you switch. Two choices do not always mean a 50-50 game, and you need to take account of the fact that a bad door has been filtered away using deliberate knowledge.

The maths is simple enough and, if you write down every possible outcome, it does back this up. But it’s a tricky thing to move your brain on from the initial incorrect assumption, just because it seems so obviously correct. A lot of scientists and mathematicians fell for it, writing highly patronising and insulting letters to vos Savant in the hopes she’d admit she was wrong. And gradually, one by one, they all sent in letters of apology and contrition.

This is a counterintuitive probability problem and, if you’re struggling to get your head around it, I don’t blame you. Forcing your brain to look past the immediate obvious jump it will make is a hard thing to do, and this is one of the reasons that probability theory can be so tricky to understand.