Comment by hansvm

There are a few other nice answers here, but I think it's important to attack it from as many angles as possible.

The intuition that you're going for is that if the true rate is 60% heads and you've seen more than that then to hit 60% odds you _must_ have some extra tails _eventually_. Interestingly, that isn't actually required to make the odds work out to 60% eventually. I'll try for an intuitive explanation:

Say you've gotten 10 heads in a row but that the coin really only has a 60% chance of coming up heads.

- After 1000 extra flips you'll have 610 heads and 400 tails total on average for a 60.4% chance of heads so far.

- After 10k extra flips you'll have 6010 heads and 4000 tails for a 60.04% chance of heads so far.

- After 1M extra flips you'll have 600010 heads and 400k tails for a 60.0004% chance of heads so far.

Notice how the average percentage of heads is getting closer and closer to 60% even though the extra flips don't have _any_ bias toward tails. A temporary bias toward tails would _also_ suffice, and in much less time (some games like WoW use this for their loot tables I think), but it isn't necessary, and in the example of independent coin flips it does not happen.