Judging the Likelihood of Uggla's Hitting Streak

From SABR member Alan Reifman at The Hot Hand in Sports on August 14:

Dan Uggla's hitting streak ended at 33 games this afternoon, as his Atlanta Braves fell to the Chicago Cubs, 6-5. Much was made of Uggla's low batting average prior to the streak and how unlikely it seemingly made the streak. In my view, judging the likelihood of Uggla's hitting streak is not so simple.

Let's start with a refresher on some principles of probability. Batting average represents a player's probability of gettting a hit in any given official at-bat. Where consecutive-game hitting streaks are concerned, we're interested in the probability of a player getting at least one hit in a game. The latter will generally be a higher probability than the batting average because the player usually will have multiple official at-bats in a game.

To estimate the probability of a player getting at least one hit in a game, statisticians typically assume a number of official at-bats per game for the player and further assume independence of outcomes (i.e., that what happened on one at-bat has no effect on a later at-bat). As of the conclusion of yesterday's play, Uggla was getting 3.76 official at-bats (AB) per game (448/119). Looking at Baseball Reference's wonderful game-by-game log for Uggla this season, he had a few games (mostly prior to the streak) with 0 or 1 plate appearances, suggesting he appeared as either a late-inning defensive replacement or pinch hitter in a few games. Assuming regular starts, which would be the case well into a hitting streak, we could estimate he'd have 4 AB per game.