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by skizm 4215 days ago
The question is phrased ad-hoc. We know the host picked a goat in both scenarios. Decision time. Do we switch or no? It didn't ask: "On an infinite timeline where the host might not always pick the goat, should we switch and if so when?" Which I think is what op was trying to say.
2 comments
dllthomas 4215 days ago
It matters why the host picked the goat. (Or, more precisely, what the odds were of him doing otherwise when he picked). That we're looking at a goat doesn't tell us everything.

Again, write the simulation. Either the switch strategy will win 2/3 of the time or 1/2 of the time, depending on how you write Monty. Only one of those approaches matches the results we saw IRL - Monty never revealed the car (so far as I'm aware) - but either is consistent with many phrasings of the question, and OP correctly specified the other case above.

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skizm 4215 days ago
you keep telling me to write a simulation. It won't matter. the code is the same for both questions since they are asking the same thing.
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dllthomas 4215 days ago
That's not true, but we'll pick it up under the code jayvanguard supplied.
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Cushman 4215 days ago
No. Even in the case of a single trial, it matters.

If the host picks randomly, and the host picks a goat, and you switch, your odds of a car are 1/3, just as if you'd stuck. There's no point.

If the host always chooses a goat, and the host picks a goat, and you switch, your odds are 2/3 in favor of now having a car. You should always switch.

Intuition suggests this. Statistics proves it. If you implement this in software, you will see it happening to you in black and white, right before your eyes.

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dllthomas 4215 days ago
"If the host picks randomly, and the host picks a goat, and you switch, your odds of a car are 1/3, just as if you'd stuck. There's no point."

1/2, once you see the goat. One of 1/3, 2/3, and 1/2 beforehand, depending on what happens when Monty shows a car: whether you automatically win, automatically lose, or start over from the top, respectively.

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Cushman 4215 days ago
Right, to be clear, that's your total chance of walking away from the show with a car. Switching doesn't change it.
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dllthomas 4215 days ago
Right.
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