Let’s play a game. Feel free to answer in the comment box. If you’re familiar with the problem, don’t give it away, but if you figure it out, please do. Please don’t Google it either, I’ll put up the link tomorrow afternoon.
The Monty Hall Problem
On the old television game show Let’s Make a Deal, there was a segment where the host, Monty Hall, would present a contestant with three doors. Hidden behind one of these doors would be a prize and if the contestant guessed the right door he won the prize. A fairly simple guess game with a 1 in 3 shot of pay-off. Every now and again though, Monty would make things a little more interesting by giving the contestant an additional choice. After the contestant had picked one of the three doors, Monty would then open one of the other two unselected doors which he knew to be empty. He would then ask the contestant if they would like to remain with their first choice or switch over to the remaining unopened door.
The question is – In such a scenario would a contestant increase their chances of winning if they a) stick with their original choice, b) switch to the remaining unopened door, or c) it doesn’t make a bit of difference either way?
Update with answer:
Well, a few brave souls in the comment box answered with the intuitive yet wrong answer of ‘it doesn’t matter either way’, knowing full well that something had to be up or the question probably wouldn’t have been asked (“who was buried in Grant’s tomb” trick trick questions not withstanding), but no one got obstinate, like I was really hoping for.
The answer, as many have explained, is that you double your chances of winning if you switch. The odds are not 1/2 either way because Monty is giving you information with his action.
To explain – the chance that the prize will be behind each door is 1/3. If you pick door #1, the combined chance that the prize is behind doors #2 and #3 instead is 2/3. Now two things are important about Monty’s action – he can’t open the door you picked, and he can’t open a door with the prize. So the 1 out of 3 times the prize would be behind door #3, he will open door #2, and the 1 out of 3 times the prize will be behind door #2 he will open door #3, but the combined odds are greater that it will be behind one of those two instead of the one you picked. For the two-thirds of the time that the present ends up behind one those doors, Monty is giving you helpful information by eliminating the empty one, and it isn’t helpful only during the one-third of the time where you have guessed the right door.
The best way to illustrate this, as Sliggy mentioned in the box, is through extreme exaggeration. What if instead of three there were a billion doors you had to choose from, and the prize was behind only one? Your odds of picking the right door on one try are now 1 in a billion, which is to say zero. But now let’s say after you pick Monty Hall goes through and opens all those millions of empty doors, except for one, and asks you if you want to switch. The only way for that other one not to be the right door is if you picked the 1 in a billion door on your first shot! . . . So still think it’s 1 in 2?
I came across this problem on the evo psych discussion group in an article linked here. What interested me is the puzzle, yes, but perhaps more so the way in which people are drawn towards intuitive answers (and how that knowledge might be useful), and how accounts of this puzzle, with amusing consistency, always seem to indicate stubborn and/or angry reactions to the information (google it and see what I mean). Take this from the article:
Indeed, some individuals I have encountered are so convinced that their (faulty) reasoning is correct that when you try to explain where they are going wrong, they become passionate, sometimes angry, and occasionally even abusive.
And, just b/c we’re Gene Expression, I’ll quote Charles Murray on it too:
. . .you should know for future reference that, stuck at a boring dinner party, you can create instant chaos, and sometimes rupture long and close friendships, by introducing the Monty Hall problem.
Update #2:
Wait, I gave up too soon, it appears Burbridge refuses to submit!:
” . . . but the probability is still equal for each door. By opening a door, he gives you information about where the prize is not, but he gives you no information about where among the remaining options it is. Therefore there is no advantage in switching from your original choice.
If anyone disagrees with this, can I gamble with them?”
The title of my post has been justified. Score!! 😛
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