#date (2003, 4, 1, 6, 14, 22, 1, 91, 0)
%title "Let's make a deal" gameshow explained
I saw this on deanesmay.com today; it's pretty old, but really made me think. I always believed the 50/50 answer, but actual verification shows that that's wrong. Oops ;)
Here's how it works:
You have a 1/3 chance of picking the right door, and a 2/3 chance of picking a wrong door.
If you pick the right door (i.e. 1/3 of the time), staying will get you the prize (+1), and changing will not (0).
If you pick the wrong door (i.e. 2/3 of the time), staying will get you nothing (0), and changing will get you the prize (+1).
So, the expected chance of getting the prize if you stay is 1/3 * 1 + 2/3 * 0 = 1/3.
(Staying gets you the prize 1/3 of the time, and loses it 2/3 of the time).
And the expected chance of getting the prize if you change is 1/3 * 0 + 2/3 * 1 = 2/3.
(Changing loses you the prize 1/3 of the time, and gains it 2/3 of the time).
Thus, changing is the better strategy. The key to it is that changing your mind is better if you were wrong at first, and you are *more likely to be wrong at first*, so changing your mind is better overall.
Interesting. I totally didn't believe this, but a quick bit of verification explains it perfectly and shows me where I was wrong. Wow. I see what Dean means about wanting to verify everything, even if it *seems* obvious. Because sometimes common sense *doesn't* give you the answer. Something to remember!
(BTW if you still don't believe that switching is the right answer, here's someone's javascript form that shows it empirically. I didn't believe it until I saw that working).