## Pages

### Baseball Probability

Question One:  The likelihood a batter realizes an official at bat each time he comes to the plate is 4/5.  His batting average is 0.300.  What is the probability that he gets a hit in each plate appearance?

Go here for a discussion of problem one:
http://www.dailymathproblem.com/2013/04/batting-averages-plate-appearances-and.html

Question Two:  Two players have four plate appearances in every game.  (I will assume all plate appearances end in either a hit or an out.)

One player has a 300 batting average (BA) against both left and right handed pitching.   The other batter has a 200 batting average against left handed pitching and a 400 batting average against right handed pitching.   Over a 10 game period, half the games are against right handers and half the games are against left handers.

(I am not allowing for substitution of right handers for left handers or left handers for right handers in this problem.)

What is the likelihood that the 300 hitter has at least one hit in all 10 games?

What is the likelihood that the hitter with a 200 average against left handers and a 400 average against right handers has at least one hit in all ten games?

Go here for a discussion of problem two:
http://www.dailymathproblem.com/2013/11/hitting-streaks-and-ability-agains-both.html

Extra Credit:  Work out the answer to the problem under the assumption used in question one that each time a person comes to the plate results in an official at bat 4 out of 5 times?

Question Three: The Table below contains data on at bats, hits and batting averages for Tony Gwynn over a 16 game period at the end of 1996.  What is the average of the game batting averages for the 16 games?  What is the aggregate batting average based on all at bats in this time period?  What is the median of the game batting average?  What is the mode of the game batting averages?

How do these statistics differ regarding the information that they provide?

 Batting Performances of Tony Gwynn Over 16 Games in 1996 Game # At Bats Hits Game BA 1 5 2 0.400 2 4 1 0.250 3 4 1 0.250 4 4 1 0.250 5 4 4 1.000 6 5 1 0.200 7 3 0 0 8 2 2 1.000 9 4 2 0.500 10 3 0 0 11 4 0 0 12 4 1 0.250 13 5 3 0.600 14 4 3 0.750 15 5 1 0.200 16 4 0 0 Total 64 22 0.347

Go here for a discussion of problem three:
http://www.dailymathproblem.com/2013/03/two-averages-median-and-mode-for.html

Question Four:  A previous post analyzed the likelihood that a hitter would have a 10-game winning streaks.   The post allowed for the hitters batting averages to differ against right handed pitchers from the batting average against left handed pitchers.  However, the batting average against all left handed pitching was independent and identically distributed and the batting average against all right handed pitching was independent and identically distributed.

Previous Post on Hitting Streaks:

How does the assumption that batting outcomes are independent and identically distributed affect the probability that a hitter has a 10-game hitting streak?

Do you anticipate that the actual likelihood of a hitting streak will be lower or higher than the estimate obtained under the assumption that outcome of at-bats and outcomes across games are iid and binomially distributed?

Go here for a discussion of how the iid assumption impacts forecasts:
http://www.dailymathproblem.com/2013/11/hitting-streaks-in-real-world.html

Question Five:  The table below has information on 1196 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
Question Six:  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?

Go here for a discussion of problem six:
http://www.dailymathproblem.com/2013/12/converting-odds-to-probabilities-for.html

Question Seven:  Below is some game-level information on innings pitched and runs allowed for a pitcher.   Use this data to calculate the following statistics -- (1) the players ERA, (2) the aggregate ERA, (3) the average ERA over all pitching appearances, (4) the median ERA over all pitching appearances,  and (5) the minimum and maximum ERA over the five appearances.

 Game Innings pitched runs allowed 1 3.00 1.00 2 0.33 1.00 3 1.00 2.00 4 5.00 1.00 5 3.00 0.00 Total 12.33 5.00

Discuss the limitations and advantages of each statistic.

Go here for a discussion on when to use aggregate versus appearance statistics in baseball