## Sunday, June 2, 2019

### Batting streaks and skill against both right and left handed pitching.

This problem compares a player who bats 300 against all pitchers to a player that bats 200 against left handers and 400 against right handers.

Question:  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 against all pitchers.  The other player hits 200 against left handers and 400 against right handers.

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?

Answer:  At least one hit means 1, 2, 3, or 4 hits.  It also means not 0 hits.  Since 0 hits is the complement of at least one hit we can calculate the probability of one hit from:

P(at least one hit) = 1 - P(zero hit),

The probability of zero hits is

P(zero hits) = (1 - BA)4

The chart below has the probability of zero hits for BA 200, 300, and 400.

 Batting Average P(Zero HIts) P(at least one hit) 0.2 0.41 0.59 0.3 0.24 0.76 0.4 0.13 0.87

The likelihood of a hitter with a batting average of 0.300 against both right and left handed pitching going 10 straight games with at least one hit is 0.7610   or 0.064.

The likelihood of a hitter with a batting average of 0.200 against left handed hitters and 0.400 against right handed hitters going 10 straight games with at least one hit 0.595 x 0.87is 0.036.

Both batters are 300 hitters.  However, the probability of a 10-game streak is 1.77 times larger for the batter who can high both left-handed and right-handed pitchers at 300 compared to the hitter who is 200 against left handers and 400 against right handers.