The Perplexing Monty Hall Problem and How It Undercuts Christianity

In keeping with the Bayes Theorem dose of probability theory last week, here’s a very approachable probability problem.

I first came across the fascinating Monty Hall Problem 20 years ago:

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?”

Most people think that it doesn’t matter and that there’s no benefit to switching. They’re wrong, but more on that in a moment.

Humans have a hard time with probability problems like this one. You’d think that we’d be fairly comfortable with basic probability, but apparently not.

Here’s another popular probability problem: how many people must you have in a group before it becomes more likely than not that any two of them have the same birthday?

The surprising answer is 23. In other words, imagine two football teams on the field (11 per team) and then throw in a referee, and it’s more than likely that you’ll find a shared birthday. If your mind balks at this, test it at your next large gathering.

Now, back to the Monty Hall Problem. A good way to understand problems like this is to push them to an extreme. Imagine, for example, that there are not three doors but 300. There’s still just one good prize, with the rest being goats (the bad prize).

So you pick a door—say number #274. There’s a 1/300 chance you’re right. This needs to be emphasized: you’re almost certainly wrong. Then the game show host opens 298 of the remaining doors: 1, 2, 3, and so on. He skips door #59 and your door, #274. Every open door shows a goat.

Now: should you switch? Of course you should—your initial pick is still almost surely wrong. The probabilities are 1/300 for #274 and 299/300 for #59.

Another way to look at the problem: do you want to stick with your initial door or do you want all the other doors? Switching is simply choosing all the other doors, because (thanks to the open doors) you know the only door within that set that could be the winner.

One lesson from this is that our innate understanding of probability is poor, and a corollary is that there’s a big difference between confidence and accuracy. That is, just because one’s confidence in a belief is high doesn’t mean that the belief is accurate. This little puzzle does a great job of illustrating this.

Perhaps you’ve already anticipated the connection with choosing a religion. Imagine you’ve picked your religion—religion #274, let’s say. For most people, their adoption of a religion is like picking a door in this game show. In the game show, you don’t weigh evidence before selecting your door; you pick it randomly. And most people adopt the dominant religion of their upbringing. As with the game show, the religion in which you grew up is also assigned to you at random.

Now imagine an analogous game, the Game of Religion, with Truth as host. Out of 300 doors (behind each of which is a religion), the believer picks door #274. Truth flings open door after door and we see nothing but goats. Hinduism, Sikhism, Jainism, Mormonism—all goats. As you suspected, they’re just amalgams of legend, myth, tradition, and wishful thinking.

Few of us seriously consider or even understand the religions Winti, Candomblé, Mandaeism, or the ancient religions of Central America, for example. Luckily for the believer, Truth gets around to those doors too and opens them to reveal goats.

Here’s where the analogy between the two games fails. First, Truth opens all the other doors. Only the believer’s pick, door #274, is still closed. Second, there was never a guarantee that any door contained a true religion! Since the believer likely came to his beliefs randomly, why imagine that his choice is any more likely than the others to hold anything of value?

Every believer plays the Game of Religion, and every believer believes that his religion is the one true religion, with goats behind all the hundreds of other doors. But maybe there’s a goat behind every door. And given that the lesson from the 300-door Monty Hall game is that the door you randomly picked at first is almost certainly wrong, why imagine that yours is the only religion that’s not mythology?

For reason is not just a debater’s tool
for idly refracting arguments into premises,

but a lens for bringing into focus the features of human flourishing.
— “What Is Marriage?” by Girgis, George, and Anderson

This is a modified version of a post originally published 11/28/11.

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• Josh Fornwalt

It’s not a matter of JUST probability, you also have to consider intuition… I’m not saying you shouldn’t change your mind, but having 2 options left out of 300 and doubting the change. You can test this by watching “The price is right”….

Intuit us the next lottery numbers…

• larrymotuz

Perhaps I am missing something in the Monty Hall problem. The initial chance of being right was one in three. Once a door was opened, the chance of being right became one in two. That is also the chance of being wrong if one stays with one’s first choice OR if one decides to change one’s choice.

So I see no benefit from changing one’s choice once one door has opened.

One could easily set up a random experiment to determine if those who changed their minds ‘won’ more often than those who didn’t. But, I feel there is no reason to assume they will, in all likelihood, do any better than those who don’t change their minds.

Do you have experimental or practical evidence of your claim?

• It’s all explained in the post. I think you need to read it another couple of times.

In short, after Monty opens one door, you have the option of staying with your original pick (1/3 chance of winning) or picking both of the other doors (2/3 chance).

• larrymotuz

Of course you should—your initial pick is still almost surely wrong. The probabilities are 1/300 for #274 and 299/300 for #59.

When there are only two doors left, both have the same probabilities of being the right door. The prior probability of only 1/300 for the door you chose is now 50%; as is the current probability for the other door. The door you initially chose now has a one in two chance of being the right one — and this is the case for the other door also.

I fail to see how this now differs since the prior probability is not the current probability.

So I think there is an error here in thinking about the current probabilities, since the past ones do not matter.

Another way of saying this is that each of the unopened doors has a probability of 298/300 — the self-same probability.And that this can be reduced to each having a probability of 50 per cent of being winners.

• Getting into long discussions about probability isn’t an option for me. And that’s part of the point of this puzzle: people bump up against probability all the time and yet they suck at it.

Unfortunately, you’re mistaken here. You can look up other web pages that discuss it. Wikipedia, for example. Their explanations may work better.

Again, it’s all in the post. Any more discussion from me would just be to repeat myself.

• Greg G.

The two doors do not have the same probability of having the prize. The door that they open has 0% chance of being the door. The other door has a 2/3 chance.

• larrymotuz

Looking at the Wiki article, I now agree. Marilyn Savant was right.

Fascinating.

And thank you!

• Greg G.

You have a one in three chance of picking the right door. That means there are two chances in three that the big prize is behind one of the doors you did not choose. You know that at least one of the doors you did not pick is not the grand prize. The host knows where the grand prize is so that door will not be opened before you are offered the switch. So your knowledge doesn’t change by the host proving that one of the doors does not have the big prize so it is still two chances in three that the other door is the grand prize.

• larrymotuz

Well, you may call it a ‘blind spot’, but, to me, the odds shifted to 50% that the door you chose was either right or wrong; and, the same 50% odds apply to the remaining unopened door. That’s because the odds are now 100% that the prize is behind one or the other door, whereas the odds before were only 2/3.

Opening the door change the odds, previously at 2/3 [i.e. that 2/3 chance that either one or the other of these doors was a winner/loser {1/3 each] to a 1/2 chance that either one was a winner/loser.

The prior probability of your first choice was 1/3 winner. This prior probability has no effect on outcomes if one simply stays with one’s decision {50%]; and the other door has that same probability [50%].

It is probability at time t that matters; not the lower probability at time t-1.

So I am suggesting that the mathematics of bringing in prior probabilities doe not capture the problem, and I would like to see evidence.

• Greg G.

Nope, it is as if you pick a door and the emcee offers you the choice of both the other two doors. You double your odds if you switch. You know one of the doors is a loser so it doesn’t matter if they prove that to you before or after you make your choice.

• larrymotuz

Much better. Thanks.

• MNb

“but, to me, the odds shifted ….”
Unfortunately probability calculation doesn’t care a bit how it looks to you or me.
The prior probability has been affected by the host opening a losing door.

• larrymotuz

Just read the Wiki article. Seems I’m in good company, but, as you say, wrong.

Thanks!

• Phil

Myth Busters did this experiment.

• Phil

At the start you have a 1 in 3 chance. Everyone agrees with that. But if you could swap your pick with the other two, you would take it? Why because you have more chance the prize being behind one of the two doors instead of the one you first picked. In fact a 2 in 3 chance. Simple? If you now open one of the doors and it doesn’t have a prize behind it, do you now think you should have stuck with the original door? You were never guaranteed a prize so you could open the second door and it not be there. But, the swap to a better odds of 2 in 3 is the wise choice and over many iterations you would come out tops.

• Greg G.

Right. There is a 1 in 3 chance that you picked the right door and a 2 in 3 chance that you picked the wrong door. There is a 100% chance that one of the doors you did not pick is a booby prize. The fact that Monty shows you one of the booby prize doors does not change your knowledge or the odds.

• Phil

Absolutely not so the remaining door still has a 2 in 3 chance of being correct.

• Monty is giving you the option of keeping your original door or having all the other doors.