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Does it matter whether or not the host knows? I don't see how it changes the odds in any way, unless of course he opens the door with the car (very likely with

by jiggy2011 13y ago

Does it matter whether or not the host knows? I don't see how it changes the odds in any way, unless of course he opens the door with the car (very likely with 1 million doors).

pbreit 13y ago
The host knows where the car is and so leaves it hidden. If some one shows you this with even just 10 doors, it will be immediately and patently obvious that you must switch. This is, indeed, the most effective way to explain it, imo.
ghaff 13y ago
It does. If the host doesn't know, there's a 1/3 probability that the game will end (you won't get a chance to switch) if the host picks the car. If this doesn't happen, it's 50:50 whether you keep your door or switch to the remaining unopened door.
Rudism 13y ago
If the host knows, you should switch. If he doesn't know, it doesn't matter--you have a 50/50 chance of winning either way.
ds9 13y ago
The fact that you asked this question shows what's wrong with the supposed "problem". In the usual presentation, it's a "trick question" because the person presenting it omits the crucial information that the offeror is intending to act adversely to the decsion making subject. Without that information, the interlocutor naturally assumes that the offeror opens doors randomly - in which case the fact that he knows what's behind each door is an irrelevant "red herring" and the correct conclusion is different. In the random-opening case the intuitive conclusion is right - that the agent (offeree)'s decision does not affect the odds. But if the question-poser revealed that the offeror is trying to maneuver the offeree into the result that's less valuable for the offeree, then the supposed mystery of the whole thing evaporates and the correct conclsion is obvious.
- hsitz 13y ago
  Ha, interesting. Yes, well, back when the problem was invented it was assumed that everyone was aware of the TV show "Let's Make a Deal" and how the scenario worked. The host/offeree (a man named Monty Hall) _never_ picked a door that had the prize behind it, implying that he _always_ knew where the prize was.
coldtea 13y ago
>Does it matter whether or not the host knows? I don't see how it changes the odds in any way Of course it matters. The host uses his knowledge to AVOID opening the door with the car (and thus ending the game prematurely). So he essentially gives out this information: "the door with the car is one that I, the host, haven't opened".
- jiggy2011 13y ago
  Let's say that you have 3 doors and you pick door #1. At that point the host says "Ok, I'm going to open one of the other doors at random and then give you the chance to change your choice". At this point 2 things could happen. Either he opens the door with the car in which case you will definitely switch to that door and have a 100% chance of winning. Or he opens the door with a goat. There is a 1/3 chance that you picked correctly in the first place, therefore there is a 2/3 chance that you picked wrongly, so it's safer to assume that you picked wrongly. You know that he didn't show you the car, so either you picked correctly to being with (1/3) or you picked wrongly (2/3), but if you picked wrongly then it must be behind the door that the host did not open. So you still have better odds from switching even though the host did not know.
  - Jtsummers 13y ago
    You are correct, as long as the host doesn't reveal the prize the odds of winning by switching remain better than the odds of winning with your initial selection. EDIT: And in the case that he reveals the prize your odds have gone up. I suspect (but haven't calculated) that a version where you can switch to the winning prize when it's revealed would give you much better odds of success and a bankrupt game show would result.
    - Jtsummers 13y ago
      Crunched the numbers, I'm wrong. If the host shows a goat (when the host doesn't know where the car is), switch or stay doesn't matter as you have the same odds either way. There are 4 outcomes to the selection phase of this game: 1/3 - both pick goats 1/3 - you pick the goat, host picks the car 1/3 - you pick the car, host picks the goat 0 - you pick the car, host picks the car So given the host picks a goat, switching or staying doesn't matter it's 50/50. EDIT: An aside - why we shouldn't trust our intuitions, but instead actually work out these problems.
scrabble 13y ago
Yes, it matters. The host opens every door other than the one with the car - or one with the goat if you picked the car. The host is not opening just one door.

Does it matter whether or not the host knows? I don't see how it changes the odds in any way, unless of course he opens the door with the car (very likely with 1 million doors).

pbreit 13y ago

The host knows where the car is and so leaves it hidden. If some one shows you this with even just 10 doors, it will be immediately and patently obvious that you must switch. This is, indeed, the most effective way to explain it, imo.

ghaff 13y ago

It does. If the host doesn't know, there's a 1/3 probability that the game will end (you won't get a chance to switch) if the host picks the car. If this doesn't happen, it's 50:50 whether you keep your door or switch to the remaining unopened door.

Rudism 13y ago

If the host knows, you should switch. If he doesn't know, it doesn't matter--you have a 50/50 chance of winning either way.

ds9 13y ago

The fact that you asked this question shows what's wrong with the supposed "problem". In the usual presentation, it's a "trick question" because the person presenting it omits the crucial information that the offeror is intending to act adversely to the decsion making subject. Without that information, the interlocutor naturally assumes that the offeror opens doors randomly - in which case the fact that he knows what's behind each door is an irrelevant "red herring" and the correct conclusion is different. In the random-opening case the intuitive conclusion is right - that the agent (offeree)'s decision does not affect the odds. But if the question-poser revealed that the offeror is trying to maneuver the offeree into the result that's less valuable for the offeree, then the supposed mystery of the whole thing evaporates and the correct conclsion is obvious.

hsitz 13y ago
Ha, interesting. Yes, well, back when the problem was invented it was assumed that everyone was aware of the TV show "Let's Make a Deal" and how the scenario worked. The host/offeree (a man named Monty Hall) _never_ picked a door that had the prize behind it, implying that he _always_ knew where the prize was.

coldtea 13y ago

>Does it matter whether or not the host knows? I don't see how it changes the odds in any way Of course it matters. The host uses his knowledge to AVOID opening the door with the car (and thus ending the game prematurely). So he essentially gives out this information: "the door with the car is one that I, the host, haven't opened".

jiggy2011 13y ago
Let's say that you have 3 doors and you pick door #1. At that point the host says "Ok, I'm going to open one of the other doors at random and then give you the chance to change your choice". At this point 2 things could happen. Either he opens the door with the car in which case you will definitely switch to that door and have a 100% chance of winning. Or he opens the door with a goat. There is a 1/3 chance that you picked correctly in the first place, therefore there is a 2/3 chance that you picked wrongly, so it's safer to assume that you picked wrongly. You know that he didn't show you the car, so either you picked correctly to being with (1/3) or you picked wrongly (2/3), but if you picked wrongly then it must be behind the door that the host did not open. So you still have better odds from switching even though the host did not know.
- Jtsummers 13y ago
  You are correct, as long as the host doesn't reveal the prize the odds of winning by switching remain better than the odds of winning with your initial selection. EDIT: And in the case that he reveals the prize your odds have gone up. I suspect (but haven't calculated) that a version where you can switch to the winning prize when it's revealed would give you much better odds of success and a bankrupt game show would result.
  - Jtsummers 13y ago
    Crunched the numbers, I'm wrong. If the host shows a goat (when the host doesn't know where the car is), switch or stay doesn't matter as you have the same odds either way. There are 4 outcomes to the selection phase of this game: 1/3 - both pick goats 1/3 - you pick the goat, host picks the car 1/3 - you pick the car, host picks the goat 0 - you pick the car, host picks the car So given the host picks a goat, switching or staying doesn't matter it's 50/50. EDIT: An aside - why we shouldn't trust our intuitions, but instead actually work out these problems.

scrabble 13y ago

Yes, it matters. The host opens every door other than the one with the car - or one with the goat if you picked the car. The host is not opening just one door.