Yes. That is what I’m saying. Your first sentence is incorrect. There are no longer three options once Monty opens a door. In a previous response to you I posted an example with 100 doors. Did you read it? It might help to understand what’s going on.
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Yes, I read it. At the end there are still only 2 doors that possibly have a prize. Switching means I choose one door and not switching means I choose the other. I say that’s a 1 in 2 chance of being correct.
We shall have to agree to disagree.
This isn’t a matter of opinion.
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I don’t “maintain” a position. Once again, I’m factually correct and you could verify that experimentally yourself. Probabilities are not a statement about the knowledge of the guesser - free will and choice are not part of any probability equation. Probabilities are a real tool that actually allow us to make predictions about reality. In this case it means that if we played this game for money, each of us betting on our own theory, I’d be more likely than not to take your money.
Actually, if anyone who thinks it’s 50% would like to play this game for money, let me know.
Then write your own simulation or do it with three playing cards. After 100 trials the odds that you will accidentally converge on the wrong thing (that is, that you’ll be closer to .5 than the correct 1/3 or 2/3) is about 130,000 to one.
If you mean that those of us who know the right answer will have to accept that people on the internet are wrong, you are correct.
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What’s more likely: that you are failing to understand the problem (no shame in that, it IS counterintuitive), or that everyone else who DOES understand it, and can demonstrate it experimentally using multiple different means, is wrong? FFS, show a little humility.
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Imagine the door wasn’t revealed, but a second option was presented:
Stay with your first guess of one of the three doors.
Change your guess and choose the other two doors, one of which is guaranteed to be a loser.
A blind contestant is essentially making this choice because they can’t see the revealed door. They just know that by switching their guess, they get two doors instead of one.
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I’m just not sure why the mathematician in the video is lying about it. She seemed nice enough…what’s her angle?
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She’s a double reverse shill for the people making the little simulators and posting them on their math websites.
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Excellent point. Admittedly, I’ve spent more hours being a DM then discussing conditional probability. Not by much, but still.
Ultimately being a DM is pretty much discussing conditional probability.
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On a small scale (three doors), this doesn’t make sense, and feels like it should be a 50/50 chance. You obviously wouldn’t choose the door that’s been shown, so you only have two real choices, stay with the door you’ve chosen, or switch to the remaining door in play.
Moving it to a much larger scale actually makes more sense to me. For example – let’s say the contest is to choose a specific star in the night sky, Wolf 359. There are something like 2500 stars visible on any given night, so if you’re guessing randomly, you have a 1/2500 chance of getting it right.
Monty Hall knows which one is Wolf 359, and he will eliminate all stars in the sky except the one you’ve chosen, plus one more that you can select. One of the stars in the sky is guaranteed to be Wolf 359. You only have two choices, but it’s far from a 50/50 chance.
You had a 1/2500 chance of a random guess being correct, but unless you know for sure that you’ve chosen Wolf 359 (because you know about these things), you should definitely choose the one in play because there’s a 2499/2500 chance of it actually being the correct star.
Big thanks to everyone in this thread for helping me understand this, esp. @anon50609448
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You know, a couple of years ago I made a real effort and did try comprehending Bayesian stats.
It didn’t last, though. I thought I could integrate it in my worldview. It didn’t work.
I still have my course notes, and the scripts. I could try it again and maybe aim for the feeling that I could understand it if my brain was better at doing ratios.
Maybe I should.
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Scenario 2 isn’t the Monty Hall problem, although it’s clearly psychologically what many people get hung up on.
If you came to the three doors and one was open, then, yes, you have a 50% chance of getting it right.
But that’s not what happened. When you picked a door, you blocked Monty Hall from picking that.
Monty is stuck by the rules of the game: he’s required to open a losing door, and he can’t pick your blocked door. Now, when you blocked a door at random, you’re in one of two scenarios:
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1/3 of the time you will have blocked the winning door. In this case, Monty can pick either of the remaining doors, and the other door is also a loser.
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But 2/3 of the time, Monty is required to pick the sole remaining losing door, and the other door is a winner!
So maybe if you didn’t think of it as “choosing and switching” but instead “blocking and then choosing,” this makes more sense.
What do you think? @simonize?
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There is really only one scenario. Do you pick one door out of three (the “hold” tactic), or do you pick the best prize out of two doors (the “switch” tactic). That Monty has opened one of the doors doesn’t chance the fact that by switching you are picking the best possible outcome of the remaining doors - it does nothing to change the odds and it is just the mechanism by which you get the best possible outcome of the remaining doors.
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So maybe if you didn’t think of it as “choosing and switching” but instead “ blocking and then choosing,” this makes more sense.
Hmmmm… that’s a different way of thinking about it. But what are we calculating the odds of? Not what Monty can do. How is scenario 2 not the Monty Hall problem? The only way I can think of is that you have to choose to switch before Monty opens the door without a prize.
Because you have prior knowledge that your first guess was most likely wrong. You don’t approach the second choice blankly.
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The trick is that you know he will always open a door without a prize, that’s part of the rules.
You are always picking between one door or two.
Two doors is two chances to win.
It would be the same if he just asked you to switch to two doors instead, opening the door is just theatrics.
The probabilities don’t change when you decide. It’s always one door or two. The prize can’t move so there is no reset.
One in three you picked right, two in three you picked wrong, so switch.
If it was 100 doors, he would open 98. You pick one out of 100 in your first pick then get told, if you picked wrong, which you probably did, then this is the right door. Do you want to switch?
The closed door is always the prize except when you choose correctly the first time.
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Because you have prior knowledge that your first guess was most likely wrong. You don’t approach the second choice blankly.
But when I’m asked if I want to switch or not, he’s already shown that door 3 does not have the prize. That is pretty clearly “3 doors, 1 is open and clearly has no prize. 1 in 2 chance”. I’m now picking between 2 doors, not the original 3 (or 100).
I’m not understanding how door 2 is suddenly having a 66% chance of being the prize if initially all doors had the same chance of being the prize.
But think about the same game with 100 doors does your gut still tell you is a 50 50 toss up between your first pick and the one other remaining door?
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