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fearmeforiampinkRight. If you already know the answer to the Monty hall problem, then feel free to read all of this, click through all the spoiler text, etc. If you don't, then I would like you to to toss a coin, roll a dice, or otherwise randomly pick between OPTION ONE and OPTION TWO below. Then answer the poll (or if you're here via Facebook/Twitter and can't do polls, answer on my facebook post about this, or post it as a comment using your Facebook/Twitter login).
Then, and only then, read the explanation area.
Option One:
You are on a game show, you have reached the final round. You are hoping to win a car. There are ten doors. Behind one of them is a car. Behind each of the other nine is a goat. (But you don't get to win the goat, it's just a signifier of no car). The car and goats are allocated randomly. You cannot smell/hear/otherwise detect through the doors.
You pick one. The host of the game show then opens eight of the other nine doors, and behind every one he opens, there is a goat (there is always only goats shown at this stage; he knows where the car is). Then he asks you if you would like to switch from the door you originally picked, to the one remaining door he hasn't opened..
Do you become more likely to win the car, if you switch?
If 'Yes' pick A in the poll below. If 'No' pick B in the poll below.
Option One:
You are on a game show, you have reached the final round. You are hoping to win a car. There are three doors. Behind one of them is a car. Behind both the other two are goats. (But you don't get to win the goat, it's just a signifier of no car). The car and goats are allocated randomly. You cannot smell/hear/otherwise detect through the doors.
You pick one. The host of the game show then opens one of the other two doors, and behind it there is a goat (there is always a goat shown at this stage; he knows where the car is). Then he asks you if you would like to switch from the door you originally picked, to the one remaining door he hasn't opened..
Do you become more likely to win the car, if you switch?
If 'Yes' pick C in the poll below. If 'No' pick D in the poll below.
[Poll #1881834]
Explanation
Wikipedia goes into more detail here; the 'simple solutions' section is especially good for making it more intuitive.
The answer is always 'Yes' (A or C). The host knows where the car is and where the goats are. You initially have an a one in (number of doors) chance of randomly picking the right door. The host opens all the doors other than the one you chose, and one other. This has not changed the probability that the door you chose is correct. But it has collapsed the possibility of the other doors into a single door, which must therefore contain the rest of the probability of success. One intuitive way of looking at it is that because there is effectively nothing behind any door that the host opens, he has effectively combined those door(s) with the final remaining unchosen door.
Typically, this is presented with a three door situation, and in that instance, most people think that they've got an equal chance whether they switch or not. Reading the wikipedia page on it mentioned the idea of turning it into an 1000 door problem, to make it easier to see the benefits of switching. I was curious, so decided to do a slightly scaled back version of that, and test it against the classical form. Another good way of presenting it is this table, which assumes you're always picking Door 1.
It's a weird problem; even when given time to try it repeatedly, humans often still don't latch onto the correct solution. Pidgeons do better at this than we do. When the problem was first explained, ten thousand people wrote in saying that it was wrong, that switching wasn't better. A thousand of those people had PhDs.
I'm curious to see how the numbers go. Since I've probably not got enough people actively reading this to get that great a number of people into the split, I'd encourage people to share this along, to see what comes out of it.
Then, and only then, read the explanation area.
You are on a game show, you have reached the final round. You are hoping to win a car. There are ten doors. Behind one of them is a car. Behind each of the other nine is a goat. (But you don't get to win the goat, it's just a signifier of no car). The car and goats are allocated randomly. You cannot smell/hear/otherwise detect through the doors.
You pick one. The host of the game show then opens eight of the other nine doors, and behind every one he opens, there is a goat (there is always only goats shown at this stage; he knows where the car is). Then he asks you if you would like to switch from the door you originally picked, to the one remaining door he hasn't opened..
Do you become more likely to win the car, if you switch?
If 'Yes' pick A in the poll below. If 'No' pick B in the poll below.
You are on a game show, you have reached the final round. You are hoping to win a car. There are three doors. Behind one of them is a car. Behind both the other two are goats. (But you don't get to win the goat, it's just a signifier of no car). The car and goats are allocated randomly. You cannot smell/hear/otherwise detect through the doors.
You pick one. The host of the game show then opens one of the other two doors, and behind it there is a goat (there is always a goat shown at this stage; he knows where the car is). Then he asks you if you would like to switch from the door you originally picked, to the one remaining door he hasn't opened..
Do you become more likely to win the car, if you switch?
If 'Yes' pick C in the poll below. If 'No' pick D in the poll below.
[Poll #1881834]
Wikipedia goes into more detail here; the 'simple solutions' section is especially good for making it more intuitive.
The answer is always 'Yes' (A or C). The host knows where the car is and where the goats are. You initially have an a one in (number of doors) chance of randomly picking the right door. The host opens all the doors other than the one you chose, and one other. This has not changed the probability that the door you chose is correct. But it has collapsed the possibility of the other doors into a single door, which must therefore contain the rest of the probability of success. One intuitive way of looking at it is that because there is effectively nothing behind any door that the host opens, he has effectively combined those door(s) with the final remaining unchosen door.
Typically, this is presented with a three door situation, and in that instance, most people think that they've got an equal chance whether they switch or not. Reading the wikipedia page on it mentioned the idea of turning it into an 1000 door problem, to make it easier to see the benefits of switching. I was curious, so decided to do a slightly scaled back version of that, and test it against the classical form. Another good way of presenting it is this table, which assumes you're always picking Door 1.
It's a weird problem; even when given time to try it repeatedly, humans often still don't latch onto the correct solution. Pidgeons do better at this than we do. When the problem was first explained, ten thousand people wrote in saying that it was wrong, that switching wasn't better. A thousand of those people had PhDs.
I'm curious to see how the numbers go. Since I've probably not got enough people actively reading this to get that great a number of people into the split, I'd encourage people to share this along, to see what comes out of it.
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no subject
Date: 2012-11-29 08:39 pm (UTC)no subject
Date: 2012-11-30 12:14 pm (UTC)Looking at the current stats (which you can do by clicking on the poll link, then clicking 'view results), they do support the thing I was checking; on the three door one it's 50/50 right and wrong here (though with a couple of extra right answers on Facebook), whereas it's 5 right, one wrong on the ten door version.
no subject
Date: 2012-11-30 01:56 pm (UTC)It definitely took me a long time of reading it, thinking it through, and getting C to explain it to me to accept the solution to the problem. I think your 10 door version would have been easier to get.
Actually, that's incredibly perceptive of you that switching up the way of framing the problem makes such a difference to understanding. If it's not been done before it would probably make an interesting paper. :)