more prisoners and hats

I though a bit about what would change if a bit of information can flow.

So assume the same situation as described in the other thread. Only change: One of the prisoners can make his guess public, before the others have to choose. Also, 99 of the 100 prisoners have to guess correctly. What success percentage does the best strategy get?

This is a lot easier to solve than the original question. I still was surprised by the result.

The first prisoner will announce the parity of the hats that he can see. Then all the others can deduce their own color.