## Monday, February 5, 2007

### Brainteaser

First off, a little brainteaser to exercise the rational thinking skills:

You're a contestant on "Let's Make A Deal", a game involving three doors, behind one of which is \$1 million, and the other two doors contain junk prizes. You pick a door, any door.

Monty (the game show host) then says, "OK, now let's do what we do in all these games: I'll open one of the doors I know to contain a junk prize." He does so. "Now, player, you have a choice: do you switch to the door you didn't pick, or stay with your original choice?"

Which do you pick: Switch or Stay?

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The answer is surprising, and you most likely got it wrong.

You probably thought that the probability of your door being right on the first try is not changed by showing us one of the junk prizes, so you chose to stay. But you'd be wrong.

To analyze a bit: the probability of you choosing the real prize on the original try is 1/3, while the probability of getting a junk prize is 2/3. This is important. PW will be taken to mean the Probability of Winning.

You make the choice. There are two possibilities: (1) You chose the right door, and (2) you chose a junk prize.
(1) Right Door: If we switch(1SW), we will lose the prize(PW: 0). If we stay(1ST), we win (PW:1).
(2) Wrong Door: If we switch(2SW), we will win the prize(PW:1). If we stay(2ST), we lose (PW: 0).
So then, looking at these probabilities:
PW(1SW): 1/3 * 0 = 0
PW(1ST): 1/3 * 1 = 1/3
PW(2SW): 2/3 * 1 = 2/3
PW(2ST): 2/3 * 0 = 0

PW(Switching Strategy): 0 + 2/3 = 2/3
PW(Staying Strategy): 0 + 1/3 = 1/3

So we can see that, regardless of which door we pick, if we choose to stay, we will only win 1/3 of the time, whereas, if we choose to switch, we will win 2/3 of the time. Mind-bending! If you chose correctly, kudos to you! You probably heard this before. If not, try to remember how you feel now, and how difficult it is to say "I was wrong." Try to remember this the next time evidence comes up against a deeply-held belief of yours. Let this experience inform you and help you realize the biases that exist in all our minds that make us fight so hard against arguments that disagree with our beliefs. I have another, similarly controversial riddle in mind for later.