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Comment by cupofpython | Hacker News Reader

if it's opened randomly then it doesn't matter, it is 50-50. You have to account for the situation where the host might reveal the prize in his random pick, and by not doing so you learn that your choice is more likely to be the prize than initially thought.

If you picked the prize P(A = 1/3), then the random open is going to be Goat P(1). if you didnt pick the prize P(B = 2/3) then the random open is going to be Goat P(1/2) or Prize P(1/2)

P(Goat reveal) happens (1/3)(1) + (2/3)(1/2) = 2/3 times

P(Prize reveal) happens (1/3)(0) + (2/3)(1/2) = 1/3 times

P(you picked prize, given goat reveal) = (1/3)(1) / (2/3) = (1/2)

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You don't have to account for the chance of him revealing the prize because he does not reveal the prize, he reveals the goat. So switch because it doubles your odds. Or don't, you're a free person. But it would be foolish.

If he's guaranteed to reveal the goat then he knows which door has the prize. This situation he isn't guaranteed to reveal a goat, he just did it this time by chance.

It doesn't matter if he's guaranteed to reveal the goat or not. He did reveal the goat, that's the scenario that's been described. So given that he did reveal the goat, should you switch? The answer is yes, from the information given, you should switch.

Everything else is overthinking.

Whether he reveals the goat by luck or because he knew which door had the prize is the entire point of the problem.

No, it's not. The point of the problem is that it's better to switch than not given the scenario presented. Whether he knows or not doesn't actually change anything, he does reveal a goat. Whether it was luck or deliberate makes no difference.

Switching doesn't change where the prize is more likely to be. That's not how the Monty Hall Problem works. It's the host giving you information about what's happening behind the scenes, by choosing a door to open and you knowing how the host operates.

Whether you end up switching or not, the other opened door is 2/3 likely to contain the prize and the door you picked originally has a 1/3. Whether you switch or not, doesn't change the odds.

>It doesn't matter if he's guaranteed to reveal the goat or not

Yes, it does. this is the entire point behind the popularity of the Monty Hall problem. It is counter-intuitive and feels like overthinking, but it is truth.

I described the situation pretty clearly (imo). I did not need to think about it much at all, though it did take some thought to describe it clearly.

I can tell you, without much thought, having been trained thoroughly in probability and statistics, it is 50-50 if a door is opened randomly to reveal a Goat. You can switch if it makes you feel better - because it is 50-50 so you wont be hurting yourself.

https://en.wikipedia.org/wiki/Conditional_probability

there are other examples that make it more clear why condition matters.

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