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
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