That’s not how probability works. You always update your probabilities based on the knowns. Your initial probability was 1 in 3 of guessing correctly. After Monty opens a door to an empty prize, your probability is 1 in 2, not 2 in 3.
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Nope, that’s not how the math works. After Monty opens a door to an empty prize, your probability remains 1/3 if you stay with your original pick, and 2/3 if you switch.
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That’s only if you don’t have the chance to switch. If you have the chance to switch, you’re sampling from 2 options, not 3.
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First of all, the opportunity to switch is what the MHP is all about: how switching greatly increases one’s probability of guessing correctly.
Second, even if the opportunity to switch is denied, the fact remains that the original guess has a 1/3 probability of being correct. It doesn’t become 50/50 if other doors are opened before the winning option or not.
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navel gazing Philosophy go!
Where is this probability located and is it a thing? ; )
Here are a few different ways to think about it, if the million door version doesn’t convince you.
First, map out all scenarios. The car is behind door A. There are three doors, you can switch or stay. That’s only six different possibilities:
| Door Choice | Stay/Switch | Outcome |
|---|---|---|
| A | Stay | Win |
| B | Stay | Lose |
| C | Stay | Lose |
| A | Switch | Lose |
| B | Switch | Win |
| C | Switch | Win |
Switch wins 2 in three, stay wins 1 in three.
Alternatively let’s say you are committed to a switching strategy. Now there are nine scenarios (car behind A, B, C / you choose A, B, C). You can’t arrive at a 50% probability to win by winning some fraction of nine scenarios.
Another way to understand it is that it doesn’t matter that they opened a door. You pick door A, then you are given the choice of staying with door A or switching to the better of doors B and C (they already showed you the worse of B and C, which is irrelevant to you choice).
But if you really want to know how it works, get ten playing cards and a friend. Pick one card as the winner, and do some 10-door Monty Hall trials yourself. You’ll believe pretty quickly that switching is better.
That’s another good way to explain that there is something other than a 50/50 probability going on.
Yes, you aren’t switching to door 2, you are switching the whichever of door 2 or 3 has the prize, if any. Clearly the probabilities that door 2 or 3 has the prize is 2/3.
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I think people have a hard time separating the probability of the prize distribution (1/3 per door…fixed at the point of hiding) form their chance of guessing the correct door.
Your first guess has a 1/3 chance of being correct because you had no additional knowledge at the time. That probability can never be altered after the fact by any amount of knowledge short of revealing the prize. Making a new choice based on new information definitely changes the odds though.
Imagine a slightly different scenario: It proceeds normally except instead of revealing a door the host allows the contestant to switch not to a single door, but all remaining doors. Obviously the chance of winning has become 2/3 for the person who switches. and remains 1/3 for the person who doesn’t. This is equivalent to the ‘reveal a losing door and switch’ scenario.
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Three doors. Each has a 1/3 chance of containing the prize. In other words, there is a 2/3 chance it does not. Or a 2/3 chance that the prize is behind one of the other two doors.
You pick a door. Monty asks you if you would like to trade your one door for both of the others. Of course you do because there’s a 2/3 chance there.
Let’s say that Monty opens one first and shows that it’s empty. That means nothing. Your odds are still better with those two doors (or the remaining closed one) even after you’ve had the proof that one is empty. You already knew one of them was going to be empty.
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It’s amazing to me that there are still people arguing with 100% certainty that the odds are 50/50 after the reveal, despite mathematicians and statisticians having settled the issue decades ago. Some of Y’all are starting to sound like climate and evolution deniers.
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I think understanding that it means nothing is kind of the crux of the matter. The probability that an incorrect door gets revealed is 100%. It provides new information about where the car is, but no new information about whether you already picked the right door - an incorrect door will be opened regardless - so the probability that you picked the right door remains the same, 1/3.
I mean, check this out:
I can understand people struggling to get their minds past their incorrect intuition, but the correct answer is the correct answer and there is no point in arguing.
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IKR? I mean, you can demonstrate it yourself, any number of ways – it’s really not THAT difficult to understand.
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My father wouldn’t believe it was 1/3 so my brother got three cards and played the game. My father stayed every time to see if it was right about half or about 1/3. After 17 times my dad hadn’t won once. As hilarious as it was unlikely.
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Thank you for this! Lightbulb on!
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To be honest, I had a lot of trouble accepting the Monty Hall problem, statistically, until I ran a simulator through a few thousand tries. It really does end up about 2/3 in favor of switching. Like Humbabella says, the right answer is the right answer. You can try to reason your way through believing the odds are 50/50 but they’re not. It’s not a complicated problem, and the solution has been found.
Now, the real life Monty Hall messing with the contestants by only opening a non-winning door in certain situations completely changes the problem. (Or, as is the case, never opening a door and not offering this scenario!) In order for the stats to work it has to be opened every time, regardless of whether the initial pick was a “winner.”
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That’s what clinched it for me, too!
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I say the problem is WHEN you have to make the decision of switching. If I have to decide to switch up front, then switching gets me 2 out of 3 chance. If I have to decide to switch after 1 selection is eliminated, it’s a 1 out of 2 chance.
You have a 2/3 chance by switching and a 1/3 chance by remaining. It’s not 50/50 before or after a door is revealed.
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Speaking of which, it’s amazing how hard a time people have accepting that 0.999… = 1. 
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There’s this weird idea that somehow you and your choice matter to the whole thing. Let’s imagine that when you come in, you are told there is a car behind door A, a goat behind doors B and C. Then you roll randomly to see which door you get. After you roll randomly to see which door you get they open a losing door that wasn’t the one you picked. Then you get whatever door you picked, no opportunity to switch. What are your odds of winning? 1/3 = the chance you rolled door A in the first place. Notice that part about opening a losing door made absolutely no difference to anything.
Unless you believe that choosing to stay at the moment after the door was revealed is somehow different than deciding you are going to stay when you woke up that morning.
I’ll admit to struggling with that one at one point in my life. I remember a big part of a math class being incredulous that anything to the power of zero is one.
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Calculating probabilities is an odd bit of maths. Some people can be good at maths but don’t get it, while others find it really easy.
So here’s what we do.
Chose one of the three doors in advance. Roll a die: 1,2 = left 3,4 = middle, 5,6 = right. Now any of Monty Hall’s head games will not work. He can put the prize in the middle if people tend to prefer the outside ones, but you have chosen at random. So, you have exactly a 1/3 chance of having picked the prize.
The other 2 doors have 2/3 of a chance together of having a prize. Monty Hall opens a door and reveals no prize, so all the remaining 2/3 of a chance lies behind the other door. There is no magic leaking of probability into the door you chose, so it has to be there. So you double your chances from 1/3 to 2/3 by chasing the other door.
There is another trick here: there is a common instinct to hang onto what is ‘yours’ and to distrust swaps. ‘Insurance’ in poker is a sucker bet for people who think the money on the table they have staked is somehow still ‘theirs’. It’s gone. If you are good at poker it may come back and bring friends.
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