Symmetry is often thought of as something that applies to geometry, but of course it can be found in algebra or even probability.

The 2007 AMC8 gave us a deck of cards with a red A, B, C, D and a green A, B, C, D, and winning hands of two cards were two of the same color or two of the same letter.

What is the probability of a winning hand?

We can do this problem the hard way: there are 8 choose 2 = 28 hands, there are 4 choose 2 = 6 red pairs, 4 choose 2 = 6 green pairs, and 4 same-letter pairs, so that makes 16/28 = 4/7.

But we can also use symmetry.  Your first card can be anything.  Whatever it is, there are 3 cards of the same color and 1 of the same letter out of the remaining 7 cards, so you have a 4/7 chance of winning on your second pick.  Because the given cards and the rules for winning don’t distinguish between the two colors or the four letters, every first card has this same chance of winning.  That’s symmetry!

1. I just saw a fun variation on exactly this idea: you pick three SET cards at random. What is the probability that they form a set?

2. Dan, that’s a great illustration of symmetry at work. That problem, and a sequence of more difficult ones, is often organized by Brian Conrey at our Math Teachers’ Circle: The Game of Set. I think work building from that even led to a published paper somewhere!