3 ms·

> Specifically, whether there was a chance that he might not have. That is false. Him opening the car is outside the constraints of the premise of the thought

by basch 6y ago

> Specifically, whether there was a chance that he might not have. That is false. Him opening the car is outside the constraints of the premise of the thought experiment. He can't not open a goat. The chance of him not opening a goat is irrelevant, and the probability is 0%.

dllthomas 6y ago
So in a new, different, and entirely unrelated thought experiment: You pick a door. Monty picks another door at random - he has clearly committed to doing so, flips a coin in front of you or whatever. God only knows what would have happened if he'd revealed a car, but this time he didn't. Having found yourself in that situation, what are your odds if you switch doors? If you simulate this procedure, you'll find that the number of outcomes where "switch" wins is about 1/3 of the total games; "stick" wins in about 1/3 of the total games; and 1/3 of the total games Monty revealed the car. When Monty does pick the goat reliably, as in the Monty Hall Problem as it was intended to be understood, Monty is doing the work to convert those "revealed" games into wins for "switch". If you disagree, I'll happily take your money.
- edanm 6y ago
  I'm sorry, I understand the difference you're implying, but can't figure out how it matters in practice. How can I even differentiate if Monty opens the door to reveal a goat by knowledge, or by chance? I mean that literally - how can I write this into a simulation to test that there's a difference?
  - dllthomas 6y ago
    It depends on the context. If it's a riddle, you could see if it's made clear by the wording, interrogate the person asking the riddle, or state the assumption you're making in your answer. If you find yourself actually in the situation, you need to make your best guess from what information is available. As for simulation, write a function that handles a single play through of the game and returns the outcome if you switch. Make Monty's strategy a parameter. Remember that "Monty revealed the car" is a separate outcome from "win" or "lose". Run that function a whole bunch of times, counting the outcomes. Compare the ratio between wins and losses, at various choices of strategy. To really drive home the point that strategy can matter, consider another possibility: maybe this season the studio executives have decided that cutting costs is more important than having an interesting game show, and Monty's strategy is now "reveal the car if you haven't already chosen it". Monty has revealed a goat. Is there still a 2/3 chance that you didn't pick the car?

by basch 6y ago

dllthomas 6y ago

So in a new, different, and entirely unrelated thought experiment: You pick a door. Monty picks another door at random - he has clearly committed to doing so, flips a coin in front of you or whatever. God only knows what would have happened if he'd revealed a car, but this time he didn't. Having found yourself in that situation, what are your odds if you switch doors? If you simulate this procedure, you'll find that the number of outcomes where "switch" wins is about 1/3 of the total games; "stick" wins in about 1/3 of the total games; and 1/3 of the total games Monty revealed the car. When Monty does pick the goat reliably, as in the Monty Hall Problem as it was intended to be understood, Monty is doing the work to convert those "revealed" games into wins for "switch". If you disagree, I'll happily take your money.

edanm 6y ago
I'm sorry, I understand the difference you're implying, but can't figure out how it matters in practice. How can I even differentiate if Monty opens the door to reveal a goat by knowledge, or by chance? I mean that literally - how can I write this into a simulation to test that there's a difference?
- dllthomas 6y ago
  It depends on the context. If it's a riddle, you could see if it's made clear by the wording, interrogate the person asking the riddle, or state the assumption you're making in your answer. If you find yourself actually in the situation, you need to make your best guess from what information is available. As for simulation, write a function that handles a single play through of the game and returns the outcome if you switch. Make Monty's strategy a parameter. Remember that "Monty revealed the car" is a separate outcome from "win" or "lose". Run that function a whole bunch of times, counting the outcomes. Compare the ratio between wins and losses, at various choices of strategy. To really drive home the point that strategy can matter, consider another possibility: maybe this season the studio executives have decided that cutting costs is more important than having an interesting game show, and Monty's strategy is now "reveal the car if you haven't already chosen it". Monty has revealed a goat. Is there still a 2/3 chance that you didn't pick the car?