The Monty Hall Problem
This problem originated when it was sent in to Parade Magazine and was published in the column of Marilyn vos Savant on September 9, 1990.
Savant was touted as the person with the highest I.Q. in Guinness Book of World Records, and while the actual value of her I.Q. is in dispute (as are all I.Q. values), I think we can stipulate that this broad’s pretty goddamned smart.
The question is based on the old game show, Let’s Make A Deal, whose host was named Monty Hall. It goes like this:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
So basically, you’re given a choice between three doors. Two goats and one car. The host opens a door you did not pick and shows you a goat. There are two doors left, the one you picked and the one you didn’t. One has a goat behind it, the other has a car. The host then asks if you want to change your pick. What do you think?
It’s a 50%-50% chance right?
Actually, it’s not. If you change your pick you actually improve your odds of winning from ½ to 2/3.
Savant got a shitload of letters from professors all over the place claiming she was an idiot. Of course, in the end, she was right.
You cannot ignore the past here like you can with a coin flip. You originally had a 1/3 chance of winning, but by switching your choice you improve to 2/3 chance to win.
The contestant should choose to switch to the remaining door. The chance of winning the car is doubled when the player switches to another door rather than sticking with the original choice. The reason for this is that to win the car by sticking with the original choice, the player must choose the door with the car first, and the probability of initially choosing the car is one in three. Whereas, to win the car by switching, the player must originally choose a door with a goat first, and the probability of choosing a goat door first is two in three.
If you’re still confused, and it took a while for it to sink in for me, the solutions and aids to understanding can be found here.
I'm going bullshit with this one, because my week started out awesom and has gone to completely depressing. And you people thought I was a contrarian before.
Regardless of what your original choice is, you're going to be offered the chance to switch. So you start out with a 50/50 chance anyways. What you choose to begin with doesn't matter a fuck bit at all, because there's always going to be a goat that wasn't chosen.
See, we forget that Let's Make a Deal, although it is a game of odds, is a game of odds that makes money through advertising dollars. No suspense, no cliffhanger; no viewers watching shticky adds for denture adhesive and auto insurance. The odds never really play into the first decision, because regardless of the player's choice, there's going to be a goat showing up along with a drum roll and the million dollar question: Would you like to see what's behind door number 2?
Obviously, I can't say this lady is dumb. I'm just calling shenanigans on the whole deal:
The first decision has no effect on the second decision because there never really are two decisions. The first decision is simple a ploy with a null option built in. You know, gotta make it to that second commercial break; might as well throw in that default cliffhanger.
God, I need someone to dictate my commentry to.
Backwards all words and G, shit.
Do the math, Einstein.
Paul - I've seen a LOT of discussion on this topic. Your explanation seems to be the easiest & simplest.