A fair coin is tossed repeatedly. What is the probability that three heads in a row will occur before a tail followed by two heads?

The probability that you can get N heads in a row would be:

Let p be the probability of flipping a heads. Let x be number of flips needed to achieve h consecutive heads. The solution is E (x) = (1 - p^h) / (p^h (1-p))

This expression may be derived as follows. The probability of being successful immediately is p^r. However, one might get a tails immediately. In that case, the number of flips needed is 1+E (x) (one flip has been used and we are back to the original position). We might get a heads and then a tails. In this case two flips have been used and we are back to the original position. Continue this up to h-1 heads followed by a tails in which case h flips have been used and we are back to the original position.