46 1. Streaks

3. Can you describe a win-loss sequence for a baseball team

that you would say exhibits streakiness, yet for which the

parameter Black is very small?

4. Suppose that a .300 hitter averages four at-bats over a six-

game stretch. Using the arithmetic-geometric inequality,

prove that the probability he has at least one hit in each

of the six games is maximized in the case that he has ex-

actly four at-bats in each game.

5.2. Basketball. The most widely referenced article on streaks in

sports is undoubtedly the paper by Gilovich, Vallone, and Tver-

sky [16]. This article examines whether or not there is auto-correlation

in the shooting sequences of basketball players. The authors surveyed

fans, collected data from games at both the professional and college

level, and carried out a set of experiments with amateur subjects. We

will give some of their most intriguing findings here.

In a survey of more than 100 basketball fans, recruited from the

student bodies of Cornell and Stanford, 91% believed that a player

was more likely to make his next shot if he had made the last two

or three shots than if he had missed the last two or three shots. In

addition, 68% said the same thing when asked about free throws. The

fans were then asked to consider a hypothetical basketball player who

has a field goal percentage of 50% (i.e. he makes about one-half of his

shots). They were asked to estimate the player’s field goal percentage

for those shots that occur after he has made a shot, and for those

shots that occur after he has missed a shot. (These numbers are our

p1 and p2.) The average value of their estimate for p1 was .61, and

for p2 it was .42. In addition, the former estimate was greater than

the latter estimate for all of the fans in the survey.

The authors compared these estimates with data obtained from

48 games involving the Philadelphia 76ers in the 1980-81 season. The

data consisted of the sequences of hits and misses for all of the field

goal attempts for all of the players on the 76ers. Among the more

interesting observations is the fact that for eight of the nine major