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.


Wine-Water Paradox

EDIT: The website auto-formats the number lines and I unfortunately don’t know how to get rid of this. You’ll just have to imagine the numbers spread out on each node of the line.
In philosophy, the Principle of Insufficient Reason (PIR) claims that if a person does not have any evidence that would lead them to believe one outcome is more likely than another, they should assign those outcomes equal probabilities. A simpler way of saying this is that in the absence of all evidence, one should assign all possible outcomes equal probability.

For instance, say I tell you that I’m going to wear either a blue, red, or brown shirt today, and want to make a bet with you in which you try to guess which shirt color it will be (assume I’ve already picked out the shirt, and can’t change it after you’ve guessed). What odds would you give to each of the three shirts? The PIR would tell you to give each 33(.333…)% odds of being chosen.

This can also be applied to continuous numbers. Say I’ve written down a number between 0 and 1. What are the odds my number is above 0.5? On a number line:

0 0.5 1

You can see that half the area is to the left of 0.5, and half is to the right, so the PIR would say the odds are 50%.

This strikes a lot of people as intuitive, and they are inclined to agree with it a priori, but the PIR is not popular among philosophers today because of several paradoxes that seem to result from it. Maybe the most famous (among philosophers who think about this stuff) is the ‘Wine-Water Paradox’:

I’m filling a pitcher with some combination of wine and water, and we know that there will be no more than 3 times of one ingredient than the other (so you can’t have 3.1 times as much wine as water, and visa versa). What are the odds of having more than 2 times as much wine as water (so the ratio of wine to water would be between 2/1 and 3/1)?

If you look at the number line:

0 1 2 3

So the odds of the ratio being between 2/1 and 3/1 are 1/(2+2/3), or approximately 37.5%.

But notice that asking if the ratio of wine to water is between 2/1 and 3/1 is equivalent to asking if the ratio of water to wine is between 1/2 and 1/3.

One the number line now:

0 1 2 3
| |
1/3 1/2

the odds become (1/6)/(2+2/3) or approximately 6.25%.

So asking the same question gives different answers according to the PIR, which cannot be.

The solution to this paradox lies in not using an ‘absolute’ number line like the ones above, but what might be called a ‘ratio’ number line:

3:1 2:1 1:1 1:2 1:3

By using this ratio line, the answer to our question becomes 1/4, either way you ask the question, thus solving the paradox.

Still this shouldn’t be construed as proof of the PIR, as there are more paradoxes out there, and even the few philosophers who do argue for it do not do so in all circumstances (e.g. you cannot use it on yes/no questions like ‘Is there a God?’).