### "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

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

(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).

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).