Your chance of being wrong in a guess is 1-1/(60* 24)

Your chance of being wrong in all 1000 guesses, assuming you guess randomly (not ensuring you never make the same guess twice) is (1-1/(60* 24))^1000

That's about .5

This is based on the assumption that you are guessing for a different person each time, so you can't increase your odds by eliminating any guesses you've already made. If you're guessing for the same person every time, you just need to guess half the possibilities. You can confirm this by realizing:

Your chance of being wrong in a guess is 1-1/(60* 24-x) where x is the number of guesses you have already made.

The product of 1-1/(60* 24-x) from x=0 to x=60* 24/2-1 is .5

Edit: hackernews formatting is weird

You could probably increase your odds significantly over time by analyzing the distribution of pins (and as others note, by making common sense observations you can also increase your odds)

60*24 = 1440 possibilities. If you guess half these numbers you would expect to crack half the pins.