Fun with numbers

Last night, while listening to Tim McCarver drone on and on about something that eventually go to Joe Dimaggio's hitting streak, I wondered what the chances were of seeing something like that again.  So I did some very simple calculations, using what seemed like decent stabs at integrating the variety of baseball history.  Came out very close to even odds.  Here are the steps I took:

1) Assume a 0.300 batting average, which translates to a 0.700 non-batting-average

2) Assuming 4 plate appearances a game, there's a (0.700)^4 = 0.2401 chance of being held hitless, or a 0.7599 chance of getting at least one hit

3) So, the chance of getting at least one hit is 56 consecutive games is (0.7599)^56, or 2.1 x 10^-7, or a 1 in 4.76 million chance.

4) 4.76 million divided by 162 games in a season, divided by 30 teams, divided by 9 batters per team, divided by 108 seasons in the World Series Era (i.e. back to 1903) and you are left with: 1.0077, or damn close to even odds.  So yes, it pretty much makes sense that someone has hit in 56 consecutive games, which means there's also a decent chance we'll see it again before I'm dead.

I should add that, as MG pointed out, if you assume a 0.285 batting average, the odds drop to about 1 in 5 against, while a 0.315 batting average makes it 5:1 in favor.