Paradoxes trick people. But no problem has deceived as many Phds, scientists, mathematicians and regular smart people; it’s called the game show host problem (or the Monty Hall paradox, named after the game show host from which this problem was derived). So, what’s the paradox? I’ll let Kevin Spacey take you through it; this is a clip from ’21-the movie’, about a bunch of MIT students trained to count cards in order to win big in Vegas.

So, what’s the deal? Shouldn’t you have an even 50-50 chance of picking the car? Right? I mean the host opened a door with a goat, so there are only two doors left and 1 has the car and the other a goat. 50-50.

No!

Why?

Think about the first door you picked. There could have been either a car behind it or a goat. Now, the host reveals another door with a goat, so:
1) if your first choice was the car, then switching to the other door gets you a goat.

2) if your first choice was a goat, then switching to the other door gets you a car!

Now, what’s the probability that your first pick, at the beginning, was the car? 1/3

And the probability it was a goat? 2/3!

There you go – look above you: By switching you have a 2/3 chance of picking a car cause there was a 2/3 chance you had initially picked a goat. If you picked a goat, switching gives you a car.

And that’s jus’ the tip.