## Wednesday, April 3, 2013

### A ten game hitting streak

Question:  The likelihood that a plate appearance by a batter will result in an official at bat is 4/5.  The batter’s batting average is 0.300.  All plate appearances are independent and identically distributed.   In each game, there are exactly five appearances at the plate.   Under these assumptions:

·      What is the likelihood that this batter gets a hit in 10 consecutive games starting at a particular date?

·      What is the likelihood that a batter gets a hit in exactly 10 consecutive games but does not get a hit in the eleventh game?

·      What is the expected number of hitting streaks lasting exactly 10 games starting at particular date if we have 300 hitters of this type?

Why might the assumption that all plate appearances are independent and identically distributed be unrealistic?  How might the deviation of this assumption from reality impact the actual number of 10 game streaks for a large number of hitters?

Answer:  First, we must calculate the likelihood of a hit in an appearance at the plate for a person with a 0.300 batting average.  We did so yesterday.  The answer is 0.240.

The probability of a hitter getting at least one hit in a game is the complement of a hitter getting no hits in a game

P(At least one hit) = 1 – P(No Hits)

The probability of no hits is 0.7605 or 0.254 so the probability of at least one hit is 1-0.254, which is 0.746.

The likelihood of 10 consecutive games with at least one hit is 0.74610 or 0.054.

The likelihood that the 10-game streak will end exactly on the eleventh day is 0.74610 x 0.254 or 0.013

The expected number of 10 game streaks from 300 hitters under these assumptions is 300x0.013=4.1

Ten game hitting streaks occur much less frequently than this analysis suggests because plate appearance are not independent and identically distributed.  The most significant reason plate appearances are not independent and identically distributed is that pitcher quality and performances vary widely.  When a pitcher gives a great game very few hitters get a hit.   As a result, even many great hitters have a large number of games where the get few if any hits.  See the previous post on this point.

Author’s note:  I am writing the second edition of my book Statistical Applications of Baseball.  Interested readers can obtain my book from Kindle.