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The fact that the show host knows what's behind the doors and never chooses a door with the prize behind it is crucial. The host leaks information about what's
by antris 4y ago
The fact that the show host knows what's behind the doors and never chooses a door with the prize behind it is crucial. The host leaks information about what's behind, as he chooses a door.
I was told about this problem and the person telling it didn't say anything about that, just that you pick a door and the host opens another one which doesn't have a prize behind it. So I thought the probabilities don't change at all because the host just picked a random door. Then everyone who heard of the problem before piled against me like I'm a dumbass, but never really explaining why I was wrong.
Yes, I'm still salty about it.
- colinmhayes 4y agoWell if the host opens a door and it never has a prize behind it they have to know which door has the prize right?
- bombcar 4y agoIt depends on if that's the scenario explained, or you have just one example (not knowing if the host will always show a goat, or if the host will sometimes show you a car either intentionally or by accident).
- antris 4y agoThe part about never having a prize behind it was not explained to me. It was just "you pick a door, host opens a door without prize". That doesn't imply that there's never a prize behind the door that the host opens.
- paxys 4y agoWhat else does "host opens a door without prize" mean?
- antris 4y agoThat the host opens a door without prize.
- gs17 4y agoThe difference is "host opens a goat door because he knows which one has the car and chooses to not open it" versus "host opens a goat door by random chance".
- paxys 4y agoFor the purpose of the problem there's no difference between the two cases. The only thing that matters is that the host (whether by knowing or by chance) opens a door with a goat behind it.
- gs17 4y agoThere is a big difference, "Ignorant Monty"/"Monty Fall" makes switching 50/50 instead of 67/33. The main change is that the door could have had the car, but didn't by chance, and we get to ignore the (one third of the) times he would have revealed a car by mistake. The original Monty never shows you the car because he always picks right, so those possibilities could never happen.
- mcphage 4y ago> That doesn't imply that there's never a prize behind the door that the host opens. The entire activity is meaningless if the host opens the door with the prize—any questions about switching or not are pointless, since obviously both closed doors are worthless. So that can’t be the interpretation of the puzzle.
- ALittleLight 4y agoI don't think the host's knowledge is important. Consider the alternative where: 1. You pick a door. 2. The host reveals a different door at random. The host has no knowledge. 3. The door the host revealed is a goat. Should you switch or stay?
- jjbinx007 4y agoThe puzzle doesn't work without the host always revealing a booby prize. In the example you gave it's just random chance. In the Monty Hall puzzle it works out that statistically you should swap.
- ALittleLight 4y agoYou should also swap in the example I gave.
- antris 4y agoNope. That version is listed here: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_behaviors https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_... "Monty Fall" or "Ignorant Monty": The host does not know what lies behind the doors, and opens one at random that happens not to reveal the car. Switching wins the car half of the time. Swapping doesn't affect the outcome, it's a 50/50 whether you swap or not.
- abirch 4y agoI would switch. The question is what happens if the door is the prize?
- tgb 4y agoThis is what I thought at first but simulating it out, it’s not true. The strategy of switching doors works 2/3rds of the time but only if you allow picking the opened door (when it has the prize). That’s easy to see since it works whenever the prize isn’t behind your first choice. Conditioning on the opened door revealing a goat gives that strategy only 50% chance. After all, the opposite situation happens 1/3rd of the time and has a 100% success rate (since you can see the car), so this other case (opened door has a goat) has to have a 50% chance to add up to the unconditioned success rate. It’s still unintuitive to me though.
- irthomasthomas 4y agoI don't think he leaks information. He invites you to switch from a game with a 1-in-3 chance to a game with a 2-in-3 chance. If you stick, you are staying in the 1-in-3 game, but if you switch then you are changing to a 2-in-3 game. That's all there is to it. Edit: I should have checked before I posted, obviously it's 2-in-3 when you switch. But it's same idea, you have better odds when you switch because then it's like you are entering a new game with only 2 doors. Still hard to believe that Erdos wouldn't believe this. He spent his life couch surfing at other mathematicians but not could convince him until he saw a monte-carlo simulation, apparently.
- dagenix 4y agoIf you switch, your odds of winning are 2/3.
- irthomasthomas 4y agoOh yes, thanks.
- jjbinx007 4y agoNo, that's not right. You need to read up on it more as it's clear you haven't truly understood the puzzle. The host knows what's behind the doors so he'll always eliminate a wrong answer. In the opening, you pick one door. That's a 1 in 3 chance of being right. So there's a 2 in 3 chance it's in either of the two remaining doors. He simply eliminates a wrong answer. Thus, 2 out of 3 times you win by switching. You win 1 in 3 times by staying with your original choice.
- crwilcox 4y agoA common way to explain this is to increase the number of doors. You have 100 doors, select one blindly, and have a 1% chance of being correct. The host removes all but one other door. Do you stay with the door you chose, a random guess of 100, or the one remaining after the host has removed all others? A door that is now statistically much more likely to be the correct choice?
- 4y ago
- valbaca 4y agoI agree. I think the way the problem is often described and explained is terrible. Why would the game show host want me to win? or want me to lose? What's his motivation behind giving me the option to switch? What knowledge does he have?
- rootlocus 4y agoYou start with a 66% chance of choosing a goat. The host opens a door and shows a goat. If you switch, you turn that 66% chance of choosing a goat into a 66% chance of choosing the prize. The fact that the host knew or didn't know what was behind the door is irrelevant. Also, if the host opened the door with the prize, what would happen then?
- etataetaet 4y agoIn your specific scenario it doesn't (I think!). However, if the host doesn't know what door the prize is behind then there is a 1/3 chance of picking the prize and you instantly loosing because both other doors are goats and you must choose one.
- niklasbuschmann 4y agoI think you're right. When you hear about the problem the first time, they just tell you "and then the moderator opens a door with a goat behind it". So you assume that the moderator usually opens a door - where in your case just happens to be a goat behind. You don't know about the show, so you assume it could've been the car, but just wasn't. So, with this assumption, it doesn't matter whether you switch. And then they tell you that the moderator always shows you a goat, but the damage is already done, your mind has made the decision that it doesn't matter.