Bags of Marbles

You have three identical bags, each containing two marbles. Bag A contains two white marbles, Bag B contains two black marbles, and Bag C contains one white and one black marble. You pick a bag at random and draw out one marble. If the marble is white, what is the probability that the other marble in the same bag is also white? Click below to see the answer.

Many people will instinctively answer 50%, or 1/2, since the marble has two possible colors, but the probability is actually 2/3 (66.67%). Why? If the first marble is white, then you know you didn't randomly select Bag B. That means that the first marble you selected has three (not two) possibilities:

1. The first marble in Bag A.
2. The second marble in Bag A.
3. The white marble in Bag C.

Only in the third case will the other marble be black, so there's a 2/3 probability that when the first marble is white, the second marble will also be white.

If you want to see how you would model this problem in Python, you can look at my solution on GitHub.

The Monk and the Mountain Path

One morning at precisely 9:00 AM a monk begins walking up a mountain path. He takes his time, stopping several times to rest along the way. He arrives at the temple at the mountain's summit at precisely 5:00 PM that evening. The next day, the monk leaves the temple at precisely 9:00 AM and makes his way back down the path. Again, he takes his time and rests at several points along the journey. He arrives back at his original starting point at precisely 5:00 PM that evening. Is there any time when the monk is in exactly the same spot on both days? Click below to see the answer.

Since the monk isn't travelling at a constant rate of speed on his two trips, it's tempting to say that there's not necessarily a time when the monk is in the same spot at the same time on both days. However, such a time and place must exist. To see why, take a look at the following plot of the two trips.

Imagine that you can grab the lines on the plot and bend them however you like, you just can't move the endpoints, and the lines must stay within the bounds of the two axes. No matter how you stretch and bend the lines, they must cross somewhere.

To think of it another way, imagine there were two monks, one at the base of the mountain and one at the temple, and they started their journeys on the same day. If they were to begin and end their trips at the same time, they would have to pass each other on the path at some point during the day.

The Pigeonhole Principle

The pigeonhole principle states that if a group of pigeons flies into a set of pigeonholes, and there are more pigeons than pigeonholes, then there must be at least one pigeonhole with two pigeons in it. More generally, if k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects. Despite its seeming simplicity (perhaps obviousness), it can be used to solve a surprising range of problems in probability, number theory, and computer science, just to name a few. See if you can use it to solve the following three problems.

1. (Warm up) A drawer contains a dozen blue socks and a dozen black socks, all unmatched. If the room is dark, how many socks do you have to take out to be sure you have a matching pair?
2. Prove that there are at least two people in Tokyo with exactly the same number of hairs on their heads.
3. Prove that if five distinct integers are selected from the numbers 1 through 8, there must be at least one pair with a sum equal to 9.

Click below to see the answers.

1. (Warm up) If you've only heard one problem involving the pigeonhole principle, it was probably the classic sock drawer problem. You only need to pick three socks to make sure you have one matching pair. When you pick two socks, you might already have a matching pair, or you might have one of each color sock. Selecting one more sock ensures that you have at least two socks of one color or the other.
2. The Tokyo hairs problem sounds like something you might be asked as a "brain teaser" interview question. If you're stuck in an interview, then the first step is to show off your estimating skills. Since we're not in that situation, we can just use Google to find out that there are about 100,000 hairs on the average human head, and that Tokyo is home to about 13.6 million people. That's more than enough people for our proof. For the sake of simplicity, let's say that 200,000 is the maximum number of hairs a person can have on their head. Then, if you select 200,001 people who happen to each have a distinct number of hairs on their heads (zero is a valid number of hairs to have on your head), you only need one more to ensure that two people have the same number of hairs. (Note: This puzzle will work with any city larger than 200,001 residents.)
3. To see how the pigeonhole principle applies to this problem, you just need to group the numbers 1 through 8 in pairs that sum to 9. {1,8}, {2,7}, {3,6}, {4,5}. Now, if I select four distinct numbers from that range, I might select one number from each of the four pairs. If I select a fifth number, then I must complete one of the pairs that sums to 9.

Coffee with Cream

Suppose you have two cups in front of you, one with precisely 8 fluid ounces of coffee, and the other with precisely 8 fluid ounces of cream. You take precisely one teaspoon of the cream and add it to your coffee. You stir it in so that it's thoroughly mixed. Then you take precisely one teaspoon of that coffee/cream mixture and put it back into the cup of cream. Does the cup of coffee have more cream in it, or does the cup of cream contain more coffee? Click below for the answer.

This question is a bit tricky. It's tempting to think that the cup of coffee contains more cream, because the teaspoon of cream added to the coffee was 100% pure, while the teaspoon of coffee added to the cream was diluted. However, it's important to remember that the total volume of each vessel changed by one teaspoon after the first transfer of fluid. The coffee/cream mixture was greater by two teaspoons than the pure cream. Before I reveal the answer, let's re-frame the question in discrete units.

Suppose that instead of liquids, our two cups contained 20 black marbles and 20 white marbles. You take 5 white marbles and thoroughly mix then in with the black marbles. Then you randomly select 5 marbles from the black/white marble mixture and place them back in the cup of white marbles. Are there more white marbles in the black cup or more black marbles in the white cup?

If you're like me, you may be tempted to treat this as a probability problem, but it isn't one. When I think about randomly drawing marbles, I want to immediately start writing a quick simulation in Python, but as you'll see, that isn't necessary. We can easily enumerate all possible outcomes in this scenario to find the answer to the question. When we draw 5 marbles from the cup with a mixture of 20 black and 5 white marbles, then place them in the cup with the other 15 white marbles, there are only 6 possible outcomes:

black	white	black/white mix	white/black mix
5	0	15/5	15/5
4	1	16/4	16/4
3	2	17/3	17/3
2	3	18/2	18/2
1	4	19/1	19/1
0	5	20/0	20/0

So the black/white ratio of the 5 marbles in the second transfer doesn't really matter. The end result is that there are always the same number of white marbles in the black cup as there are black marbles in the white cup.

The same is true of the original coffee/cream problem. The ratio of the two liquids in the teaspoon that is transferred to the cup of cream is such that you will end up with precisely the same volume of coffee in your cream as there is cream in your coffee. So the answer to the trick question posed at the beginning is "neither, they are the same."