The following two problems use baseball data to
explain two concepts – odds and probability.
Question
One:
The odds of the Boston Red Sox winning a game at home in 1996 were 1.382 while
the odds of winning on the road were 0.884. The Red Sox played 81 games
at home and 81 games on the road. What is the probability that the Red
Sox would have won a game at home and the probability that the Red Sox would
have won a game on the road? How many games did the Red Sox win at home,
on the road, and in total?
Answer: The
odds an event occurs is defined as
odds=p/(1p)
where
p is the probability that an event occurs.
The
problem for home games is solved by plugging in odds = 1.382 and then solving
for p.
1.382=p/(1p)
1.382*(1p)=p
1.382 =
(1+1.382) x p
p = 1.382/2.382
= 0.5802
The
problem for away games is solved by plugging in odds=0.884 and rearranging
0.884=p/(1p)
Solve
and get:
p =
0.884/(1+0.884) = 0.4692
To
get the number of victories at home and away multiply the respective victory
probabilities by the number of home and away games, which is 81. I get
47 home victories, 38 away victories and a total of 85 victories.
The Boston Red Sox did win 85 games in1996 coming third in their division.
Question Two: The table below has
information on 1996 regular season batting date for Cal Ripken. What
are: (1) the odds of Cal Ripken getting a hit or not getting a hit on a
specific at bat; and (2) the odds of a home run or not getting a home run on a
specific at bat?
Data
for Cal Ripken 1996 Season


Singles

111

Doubles

40

Triples

1

Home
Runs

26

At
Bats that Did not end in a hit

462

At
Bats

640

Batting
Average

0.278

Definition of odds: The odds an event
will occur is defined as
P/(1P) where P is the probability the event will occur and 1P is
the probability that the complement of the event will occur.
Answer to Example 5.2: The odds of a Cal
Ripken hit are 0.385 (0.278/(10.278)) and the odds of not obtaining a hit are
the reciprocal 2.60. Home run odds are 0.043 (0.041/0.959) while odds of not
getting a home run are the reciprocal 23.4.
Author's Note:
These problems first appeared in my book Statistical Applications of Baseball, published in 1996. It is available at a very low price on kindle.
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