[madcaptenor]

I like the idea of simulating. But not all distributions are normal! In particular there's no reason to expect the distribution of hitting times to be normal, and intuitively it seems like the distribution would be very far from normal.

I wrote a program and did ten thousand trials; the 100th largest hitting time was 5578, which is my estimate of the 99th percentile of the distribution, and thus the answer to the version of the original problem where we don't have to fix the number of coin flips ahead of time.

The median hitting time from my simulation is 152; the mean, 506, the standard deviation, 1155. Yes, the standard deviation is larger than the mean! The distribution has a very long right tail.

I'm fairly confident (in an informal sense) that the median of the true distribution is somewhere near 152, but I'm not confident about the mean or standard deviation. A lot of distributions in problems like this have tails which decay only like power laws, which makes estimating the mean and standard deviation from a sample very difficult. (I'm not saying that the distribution is a power law, though; it's hard to identify those just by looking at the data.)