[pdurrant]

Originally Posted by jbjb

Spoiler:

If you pick off a head in the first draw, and don't turn it over, you're now safe. He now has 20 heads and you've got 1. By picking and turning over the next 19, you can be sure to match his number.

/JB

Yes, I agree. Another way to look at it:

Spoiler:

You pick a coin and it's a head.

There are now 20 heads and 80 tails left. Pick 19 coins than have X heads in them. The gaoler is left with 20-X heads. You turn your coins over and now have 19-X heads. Add in the one head you first picked, and you have 20-X heads, matching the Gaoler and you go free.

Of course, if you picked a tail with that first coin, the gaoler has 21-X while you only have 19-X.

But yes, a definite 21/100 chance of freedom

So — we have a strategy that gives just over a 20% chance of freedom. Can anyone do any better?