Comment by dragonwriter

> If I’m expecting 60% of my flips to be heads, and I’ve already had 60%, isn’t it more likely that the next one will be tails?

Nope.

> I’m sure you can probably tell I know next to nothing about either maths or probability, so feel free to explain why I’m wrong.

Lots of people have explained in terms of independence, which is correct. Another way of looking at it (definitely not more correct, but maybe more compatible with the “a series should eventually match the quoted probability” thinking) is in terms of infinity:

If you are expecting 60% of results to be heads, you expect that to hold over an infinite series of flips.

If you see any finite number of heads in a row, the probability for each of the remaining flips in the infinite series to get the total to 60% is…still 60%.

No finite series of results can change the probabilities necessary to get the infinite series to turn out as expected.