Two Math Problems

Jerry Coyne, over at Why Evolution is True, poses two math problems.

The first is a case of what is called ‘sampling without replacement’:

You have four balls in a sack. Two are black and two are white. You reach in with both hands and (simultaneously) pull out two balls. What are the odds that the two balls will be different colors?

Answer: 2/3

You might intuitively think that the answer should be 1/2, but keep in mind that each hand must grab a different ball. So if your left hand grabs a black ball, there will be three balls left for your right hand to grab, and two will be white. Likewise if your left hand grabs a white ball.

The second is the famous ‘Monty Hall’ problem, which is used to illustrate the concept of condition probability:

You’re on a game show, hosted by Monty Hall. You’re presented with three doors. Behind two of the doors are goats. But behind one of the doors is a brand new car. After you pick one of the three doors (but do not yet open it), Monty Hall chooses one of the other two doors that does not have the car behind it (it is part of the rules of the game that Monty Hall always chooses a door with a goat behind it), and removes it from the game, so you now have two doors: the door you chose, and the door not removed by Monty Hall. You are then given the chance to switch doors before opening them. Should you switch?

Answer: Yes, you should switch. Switching gives a 2/3 chance of winning.

Almost everyone thinks the answer is that it doesn’t matter whether you switch or not, the odds will be 50/50 (that was my answer the first time I heard of this, too). The key lies in the fact that Monty Hall is guaranteed to remove a door with a goat behind it.

Consider another version of the problem. This time, you are presented with 100 doors. Behind one is a new car; behind the rest are goats. You choose one door at random, and Monty Hall removes 98 of the other doors, behind which are guaranteed to be goats. Now, the chance of you choosing the door with a car behind it on the first try is only 1%, so the other remaining door has a 99% chance of having the car behind it. You should obviously switch.