> > Given what we know, and since the problem's symmetric with respect to heads/tails, the chance of getting a heads is 1/2 on the first flip. I.e., we're 50% confident of getting heads when we're just about to make the first flip.
> >
> > The question is this: if we keep flipping the coin and getting heads, after which flip do we become 99% confident (or higher) that the next flip will also be heads?
>
>
> I believe the answer is: never. The odds remain the same, 50/50, no matter how many times you flip the coin.
>
> koala"had to switch to statistics after flunking calculus"mom

That would be the case if it were a fair coin, but the supposition is that it's not a fair coin. The reason it starts out as a 50/50 shot on the first toss is that, though the coin's going to give skewed results, we don't have any evidence on which way it will be skewed at first. Because of the symmetry of the situation, it's still perfectly balanced between heads and tails.

Each successive heads we observe increases the (conditional) probability of the coin being skewed towards favoring heads. [This is contrasted to the *actual* probability of getting heads, which stays constant at p. We have to be careful about which probability space we're looking at things from -- this depends on how much information about the experimental conditions we have.]

As to arriving at the solution: it's not completely trivial, but it can be done in your head with minimal calculation if you think about it the right way.