Matching Socks?

In this three-part puzzle, we need to help Dr. Bumble match his socks.

Puzzle 1

Dr. Bumble has a box of socks in his closet containing five black socks and five blue socks. He wants to be certain his socks match, but he cannot see the colors, since his closet is completely dark owing to an electricity outage. What is the smallest number of socks that Dr. Bumble needs to remove from the box in order to be certain of having two socks the same color?

Puzzle 2

Being a professor of statistics, Dr. Bumble is willing to accept a bit of uncertainty in his life, but he does want to have at least an even chance of getting two socks that match. How many socks does he need to remove from the box for this to be the case?

Puzzle 3

After he gets dressed, with matching socks, Dr. Bumble wonders how many socks he would have needed to pull out of the box to have at least an even chance of getting two socks that match if there were an infinite number of socks, half blue and half black.

Course:

Math [1]
Probability [2]

Result/Solution(s)

Solution: Matching Socks? Math Puzzle

Puzzle 1

Dr. Bumble needs to draw three socks to be certain of getting a match. The first two socks may be different, one black and one blue. But the third sock will certainly match one of the first two.

Puzzle 2

To have at least an even chance of a match, that is, a probability of 1 / 2 or greater, Dr. Bumble would still have to draw three socks. After drawing the first sock, there are four socks left of that same color and five of the other color. So the probability of a match on the second draw is 4 / 9, which is less than 1 / 2. He needs to draw a third sock to get at least an even chance of a match. And in fact, as we saw above, he is now certain of a match.

Puzzle 3

As the number of socks in the box increases, the probability of the second one matching the first increases, getting closer and closer to 1 / 2. Let’s say there were 50 black and 50 blue socks in the box instead of 5 of each. Then the probability of a match on the second draw would be 49 / 99 instead of 4 / 9. If there were 5,000,000 of each, the probability of a match would increase to 4,999,999 / 9,999,999. As the number of socks approached infinity, the chances of a match on the second draw approaches 1 / 2.

What if there were equal numbers of socks of three different colors in the box?

Probability puzzle [3]
Math Puzzle [4]