Last night we watched a movie called “21” – about a group of MIT math students who made a fortune in Las Vegas by “counting cards” at the Blackjack tables. The movie was so-so, but one scene reminded me of an old mathematical nemesis — the Monty Hall problem. In the scene, a math professor (Kevin Spacey) tells the student there is a car behind one of the three chalkboards, 1, 2, or 3. He asks the student to guess which chalkboard hides the car. Student picks board #1.
At this point, professor reveals that behind door #3 there is no car. So now we know that the car is behind either board #1 or board #2. Professor then asks student if he wants to change his guess. Student says “yes” and changes his guess to board #2, telling professor “I have a 66% chance behind board #2, but only a 33% chance behind board #1.”
This is unintuitive to me. When one choice becomes eliminated, my intuition tells me that the probability of the car behind either remaining door is 50:50, regardless if I had previously made a choice, or not.
So I created a test with my son. We did 40 trials. He guessed 1 of the 3 options, then randomly said “keep” or “change” without even knowing the choice I had eliminated. Indeed, when he changed his original choice, he was right roughly 2/3 of the time. When he did not change his choice, he was right only 1/3 of the time. (for those still puzzled, a good analysis can be found here).
Does this reveal a larger metaphor? More times in life than I care to admit, I have found myself holding on to a form of certainty that was later found to be totally unfounded. Even when presented with unassailable evidence, we often refuse to acknowledge the clarity set before us – favoring deeply ingrained “religious” certainties. And some of you are still saying “no, there are two remaining doors – it’s plainly obvious to anyone that the odds are 50/50.”