I think I may have figured it out. You have to use binomial probabilities and a calculator like this one:
http://vassarstats.net/binomialX.html
Let's say your record was 33-27. That's 60 bets. You want to know what the odds that a coin flip could achieve those results. The less likely a coin flip could achieve it, the more likely that you are a skilled bettor.
Enter:
n = total bets
k = losses
p = 0.5
Then hit Calculate.
Look at the
P: k or fewer out of n number. In this example, it's
0.259479001595
Multiply by 2 (you want both ends of the spectrum): 0.259479 x 2 =
0.518958
Therefore, there is a 51.89% chance that a coin flipper could have achieved that result or better. So there isn't good evidence that you are a skilled bettor at 33-27. It's just as likely to be chance.
Another example, record of 36-24
P =
0.077500952002
P x 2 =
0.155001904
Therefore, there is a 15.5% chance that a coin flipper could have achieved that result or better. So it's very likely that your record is due in part to skill.