28 1f a coin is flipped 25 times with eight tails occurring

28. 1f a coin is flipped 25 times with eight tails occurring, what is the probability that no run of six (or more) consecutive heads occurs?

Solution

i just ran a simple computer simulation and got 447,669 \"winners\" out of 1,081,575 cases which have exactly 8 tails occurring (out of 25 fair coin flips), so the probability is 41.39%. I looped thru all 225 = 33,554,432 possible states of 25 coin flips. There is some \"state pruning\" I could have done but didn\'t bother cuz the program already ran fast enough. Things like not looping thru the cases where the first 6 coins are already heads (binary 111111xxxxxx...) where x is a don\'t care situation (either a 1 or a 0). There are 219 = 524,288 of these so that is not much compared to about 33.5 million but if the question had been out of 30 coin tosses, then maybe that tweak would have been more beneficial.