## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## May 11, 2007

### Batting Streaks

For baseball fans, one of the highlights of the 2001 season was the home-run record set by Barry Bonds of the San Francisco Giants.

All summer long, newspapers printed charts showing how many home runs Bonds had hit and how many he would have at the end of the season if he continued hitting them at the same rate. On average, Bonds was hitting a home run once every second game. So his total was likely to be about half the number of games he played. In the end Bonds played 153 games and walloped 73 homers.

Here's an interesting question. Bonds had a 50 percent chance of hitting a homer over the course of the season, but were those his odds for any particular game as well? What if he had a hot streak? What if you went to a game when he was in the middle of his hot streak? Wouldn't the chances of his hitting a homer be higher than even-steven?

Another way to ask this question is to ask whether Bonds was hitting homers like a tossed coin falls. When you toss a fair coin, it has a 50 percent chance of coming up heads. Even if you get heads three times in a row, the next time you tossed the coin it would still be even odds it would come up heads. Was Bonds's ability to hit as random as a coin toss, or could he, by eating his Wheaties or spitting on his hands, force a long run of good luck?

For each game during last summer's baseball season, economist Paul Sommers of Middlebury College in Vermont checked whether Bonds hit one or more home runs and looked for streaks. Bonds had one stretch of 13 games in which he failed to homer. He had two stretches of six games in which he hit home runs every time.

You can get streaks when you toss a coin, too. Suppose you toss a fair coin 250 times. You will probably get two runs of six heads or more, and one run of seven heads or more. Most people are surprised by this. When they are asked to write down a long string of heads and tails that they believe is random, they rarely include four or five heads in a row, even though such runs are likely to occur.

Sommers used a statistical formula to find out whether Bonds's hot streaks were as random as home runs in a coin toss. They were. How could this be? Maybe so many factors affected his hitting—the weather, game time, the ballpark, whether the pitcher was right- or left-handed, and so on—that it wasn't really possible to predict whether he'd hit a homer in a particular gasme. All you could say is, based on past performance, his odds were about 50-50.

Somehow that sort of takes the magic out of a streak. But it doesn't mean Bonds isn't a good player. A lesser player might have as much chance of hitting a homer as a single die has to come up as a three (one chance in six).

Muse, May/June 2002, p. 35.