What is the probability that, at some point during this series of 10 flips, the difference (absolute value) between the number of heads and tails is 3 or more. Assume the coin is fair. Is it possible to find a general formula for this problem, that allows you to simply plug in the number n flips, the difference d (3 in the above problem), and the probability p that a single flip results in heads (assumed 1/2 above), and out pops the required probability? edit: This is for enjoyment, only. I can think of one way at least (using probability matrices) to get the answer for any particular set of values. I'm just interested to see what other ways people can think of. Thank you.