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?