i was randomly thinking about joe dimaggio's 56 game hitting streak and was just wondering what the odds of doing that are.

i dont know what hi batting average was but lets say it was .333

do you just take .333^56?

i think you have to factor in his plate appearances per game as well. for this argument just say on avg he gets 4 plate appearances per game. if he bats .333, in a perfect scenario his hits should come every 3 at bats but we know thats not how it always works.

If he's always a .333 hitter (makes no difference who's pitching), and he gets 4 at bats every game, and he never walks (because a walk doesn't affect batting average but also doesn't count as a hit). Then each game he has a .667^4 = .198 probability of not getting a hit, therefore a .802 probability of getting at least one hit. So his probability of getting a hit in each game of a particular 56 game stretch would be .802^56 = 4.3*10^-6.

Note that this is just for one particular pre-determined string of 56 games, not his probability of getting the streak at some point in a season. Also, we didn't start with the most realistic set of assumptions (he's going to walk sometimes especially when he's hitting well, different pitchers are different, etc).

the being walked thing i did consider i just went based on plate appearances since unless the player is intentionally walked he is given the opportunity to get a hit in every at bat.

You have to know how many games he played in his career (and somehow do a weighted batting average across the span) to find the chance that a 56-game streak happened within that series. Just raising P to the exponent of the streak as you show above, is the chance that something happens exactly on the NEXT 56 trials (as ES already pointed out). That's completely different.