The Monty Hall Problem: Why People Get It Wrong

Marilyn and the Monty Hall problem

The other day, I was mindlessly scrolling one of my social medias when I saw a post about Marilyn vos Savant and the famous incident of the Monty Hall problem in her "Ask Marilyn" column.

For those of you not familiar, the Monty Hall problem is a probability question which was originally phrased in the column as:

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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Marilyn answered that yes, if you switch, you have doubled your chances of winning.

At the time, as a supposed genius, she was absolutely excoriated for this answer. Scores of people, many of them men, including academics, mathematicians and holders of doctorates, wrote in to tell her she was wrong and the player would simply be faced with a 50/50 probability, meaning it didn't matter whether they took the offer to switch or not; they would be no better off either way. Many of these letters contained sexist comments, references or suggestions that women were clearly worse at maths than men.

Now, to cut a long story short, Marilyn was correct and it's been mathematically proven, repeatedly - on paper, in real experiments and computer simulations. You are indeed twice as likely to win if you take the switch. That fact, today, is not in any dispute whatsoever in academia. It is disputed still by some people who don't understand the problem and can't wrap their heads around why their intuitions are wrong.

Humans are notoriously bad at dealing with probability on an intuitive level and while there's certainly no excuse for the way Marilyn was treated for her correct answer, it is very understandable why so many people then and even today (as the post I saw on Facebook demonstrated) continue to get it wrong.

Why it seems like Marilyn's answer is wrong

So what's going on when people are adamant it's 50/50? The answer is in the crafty wording of the problem, which leads people to believe that the elimination of one of the options is resetting the scenario, leaving the player with a random choice between two, equally likely doors.

And it's true that if you did have a random choice between two doors, one with a prize and one without, your odds would be 50/50.

But we need to understand this isn't what's happening in the Monty Hall problem.

To help us get our head around the difference, let's look at a couple of similar scenarios that would be a 50/50 chance.

Let's also get rid of this confusing stuff about doors, cars and goats. It's just unnecessary imagery. Instead we'll just imagine three boxes; one contains a prize and the other two are empty.

This is how I'll refer to the Monty Hall problem for the remainder of the article.

Scenario 1: I eliminate a dud before you choose

I offer you three boxes. I tell you one contains a prize and two are empty.

Then, before you make any choice, I go "actually, you know what, this box here, number 3, is empty, so let's forget about that one" and I open it, show you it's empty, and chuck it away.

In this scenario, you are now choosing randomly between two boxes with a 50% chance of picking the prize.

You pick	Winning box	Outcome
1	1	Win
2	1	Lose

Scenario 2: I eliminate a box at random after you pick

I offer you three boxes. I tell you one contains a prize and two are empty. I do not know which box contains the prize.

You pick a box. After you've picked, I open one of the two boxes I have left at random. If the box I open contains the prize, no switch can be offered, you've just lost. If it's empty, I offer you the switch. Let's look at the possible outcomes that can happen here if you've picked box number 1:

You Pick	I Open	Prize In	Result Switch	Result No Switch
1	2	2	-	Lose
1	2	3	Win	Lose
1	3	3	-	Lose
1	3	2	Win	Lose
1	2	1	Lose	Win
1	3	1	Lose	Win

This truth table looks the same for the additional rows where you pick box 2 or 3 - there's no option to switch in cases I opened the box containing the prize - you just lost immediately there, so those are eliminated, leaving us with:

You Pick	I Open	Prize In	Result Switch	Result No Switch
1	2	3	Win	Lose
1	3	2	Win	Lose
1	2	1	Lose	Win
1	3	1	Lose	Win

And you can see clearly, you will win 50% of the time, regardless of whether you switch or not.

What happens in the Monty Hall problem

There are two subtle but extremely important differences between the scenario above and the problem Marilyn was given:

1. As the host of the game show, I know which box has the prize.
2. As the host of the game show, I must offer you the chance to switch.

The corollary of these two points is that when I open a box after you've made your choice, there are two rules I must obey:

1. I can't open your box.
2. I can't open the box containing the prize.

Now, 1 in 3 times you will pick the box that has the prize, which means (1) and (2) above will be the same box, so I as the host can open either of my two boxes to offer you the switch, it doesn't matter which one I pick.

But 2 in 3 times, you will pick an empty box, leaving me with one empty box and one box holding the prize.

In those 2 out of 3 runs, this means there is only one box I can open - the one box I have that both isn't the box you picked and isn't the prize.

Now look at the truth table for all the possible outcomes of this scenario:

You Pick	I Open	Prize In	Result Switch	Result No Switch
1	2 or 3	1	Lose	Win
1	3	2	Win	Lose
1	2	3	Win	Lose
2	1 or 3	2	Lose	Win
2	3	1	Win	Lose
2	1	3	Win	Lose
3	1 or 2	3	Lose	Win
3	1	2	Win	Lose
3	2	1	Win	Lose

When you switch, you win twice as often as when you don't.

Still confused?

The key to understanding this problem and its highly counter-intuitive answer is in understanding the host's choice of which box to open isn't random.

As we've seen in our various tables above, if the choice of which box to eliminate is random, either before or after you've made your choice, you are indeed left with a 50/50 chance. But that would mean the possibility of the host opening the box containing the prize and therefore ending the game before they've had a chance to offer you the switch.

The puzzle is really clever and crafty, because the wording makes it seem on one level like you're being offered a simple and equal swap - would you like to switch one box you've picked for another, random, equally likely box?

But it's those critical words - "who knows what's behind the doors" - which change everything.

When you first pick a box, you know you have a 1 in 3 chance of winning. And you know, conversely, there's a 2 in 3 chance the prize is in one of the boxes you didn't pick.

You also know, therefore, that at least one of those other two boxes must be empty, regardless of whether you've picked the prize or not.

So when the host reveals - with full knowledge of which boxes are empty - that at least one his two boxes is empty, these probabilities haven't changed.

There is still a 2 in 3 chance that the prize is in one of his 2 boxes. By pointing to one of them in particular and saying "this one is empty", you're being tricked into thinking the host has somehow reset the odds on the remaining boxes, but they haven't changed at all. The host knew that box was empty, before you picked anything. So all he's done is give you more information. Now, instead of knowing there's a 2 in 3 chance the prize is in either one of his two boxes, you know there's a 2 in 3 chance it's in the box he didn't open.

This means the question you're being asked, in effect, isn't "would you like to swap your box for this box?", it's "would you like to swap your box for both other boxes?" - and I hope this makes it obvious why accepting the switch is in your favour.

Conclusion

If, even after reading this article, you're still struggling to get to grips with why you should take the switch -- don't worry! When Marilyn published her answer, even seasoned mathematicians couldn't grasp it. Probability is hard.

I sometimes play the lottery. In the UK, the odds of one ticket winning the lottery jackpot are roughly 1 in 45 million. I once had a conversation with someone who was absolutely adamant that there was a difference between 100 in 45 million and 1 in 450,000. Their explanation was - and I quote - "if you bought 100 tickets, your odds would be 100 out of 45 million, not 1 in 450,000, otherwise it would be easy to win the lottery by just buying 100 tickets every week until you won"

I did the maths and pointed out to them that on odds of 1 in 450,000 (which yes, is identical to 100 in 45 million), to have a greater than 50% chance of winning the jackpot at least once, you would need to buy 100 tickets for both weekly draws, every week without fail, for approximately 3,000 years. In other words, don't give up the day job just yet.

Then we have the so-called Birthday Paradox, wherein it turns out you only need around 30 random people in a room to have an approximate 90% chance that two of them share the same birthday (by day and month, not year).

Or another example dealt with by Marilyn in her magazine column, known as the Boy-Girl problem, which asked:

A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "Is at least one a male?" she asks him. "Yes!" she informs you with a smile. What is the probability that the other one is a male?

If you struggled with Monty Hall, you might want to give this one a miss!

Bottom line: We humans don't handle probability well. It often works against the grain of our brain's pattern-seeking intuitions. Luckily for us, we are quite good at developing mathematical tools to help us calculate and understand probabilities, with a little effort.

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