Posted on October 20, 1963 in General Questions by MontyHall:
Howdy folks! I’m the host of a TV show that’s going to debut in a few months, but Cafe Society isn’t the place for this, because what I have is a math/probability question.
On my show, people will be given such choices as whether they want what’s in a big box or a little one. The big box may contain a refrigerator – or a pile of rotted lumber! The small one could have just a paperback book – or a diamond ring!
The person who ends up with the biggest prize will get a chance to win a brand-new car at the end of each episode! However, there will be three curtains. Behind one will be a Pontiac Tempest, Ford Galaxie, or other fine automobile. However, behind the others will be nothing but live goats or donkeys! We’re going to call these “prizes” zonks.
Now here’s where my question comes in. Let’s say the car is behind Door #2, and the contestant picks #3. To heighten the suspense (and prolong the agony), I think it would be a good idea to show him the zonk behind Door #1, and then ask him if he wants to switch his original pick to Door #2. In this case, if he does, he wins – if he sticks with his first impression, all he gets is livestock!
Some of the performers and production staff members think that once there are only two doors left, the odds will be 50-50 that the contestant picks right. However, others say that there’s a mathematical advantage to switching. We decided to test this, and played 10 sample games using each strategy. The guy who stuck with his original pick won 4 of his 10 times, while the one who switched won 6 of 10.
However, this may be just a coincidence, as we all know it’s possible to get “heads” on 10 straight tosses of a penny even though the odds of “tails” are 50-50 each time. So what we want to know is if there’s a mathematical theory that explains why the switcher was using good strategy, or was just lucky. You Dopers are a smart bunch, and I just know one of you mathematicians or physicists will come up with the answer right away, so this thread can be closed after just a couple of posts!