The following question was presented:
Time for one of those questions that you need to think about. I actually got this one from Mythbusters, so you may have seen it. Anyway, there's a gameshow where the contestant is presented with three doors behind one of which is a prize. The contestant is asked to pick a door. The host then opens one of the remaining doors that doesn't have a prize behind it and offers the contestant the option to switch doors or stick with their original choice. Should the contestant switch doors to maximize their chance of winning?
Results:
Yes - the contestant should switch doors and It doesn't matter - the odds are the same tied with 40% of the votes each from the following selection:
- Yes - the contestant should switch doors (40%)
- No - the contestant should stay with their original choice (20%)
- It doesn't matter - the odds are the same (40%)
Graphic of Results:
Analysis:
I thought most people would get this as it's a fairly simple problem. The correct answer is yes, you should switch. There's actually twice the possibility of being correct is you switch than if you stay with your original choice. The reason behind that is that there is a 1/3 chance you are correct with your initial choice - 3 doors and only one is correct. There is a 2/3 chance that the correct door is one of the remaining ones. You can look at the problem solution as getting a choice of 2 doors if you switch. It just happens that you know that one of them is incorrect (which you would do anyway if you were allowed to pick 2 doors). There is no resetting of the probabilities just because a door has been opened.
The interesting thing about the Mythbusters is that they tested two aspects of this problem. They did indeed test that it's statistically better to switch your choice and it did result in switching providing a winner on about a 2 to 1 basis. This was done by running the test multiple times and looking at the data, not just looking at the math behind the solution.
The second component they tested was whether or not people were more likely to switch or if they would stick with their original choice. It actually turned out that 100% of people stuck with their original choice. Psychologically, this is a great game because even if you know you are mathematically better off switching, something is telling you to keep with your original choice. Presumably the fear that you don't want to switch in case you were right with your original choice.
All this said, perhaps we should have asked how far you can shoot a cannonball.