Tuesday, August 6, 2019

Baseball odds and probabilities

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/(1-p)

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/(1-p)

1.382*(1-p)=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/(1-p)

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/(1-P) where P is the probability the event will occur and 1-P 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/(1-0.278)) 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.