My Question: In the derivation (provided below), the case where more than k consecutive heads occur isn't considered.
Length of the Longest run: If a coin is flipped n times, what is the probability that there is a string of k consecutive heads. (probability of heads is p)
Just considering the events to be series of K flips creates too much correlation amongst the events, so E(i) is considered to be the event that i .... i+k-1 flips are heads and i+kth flip is a tail. This makes E(i) non overlapping events, and Poisson Approximation can be used.
Consider this part of the 2nd image:
"Because there will not be a run of k heads if (and only if) N = 0, thus the preceding gives
P(no head strings of length k) = P(N = 0) = exp{−(n − k)(p^k)(1 − p) − (p^k)} "
However, I think if N=0, which means none of events E(i) occurred, that only means that there is no string of exactly K consecutive heads. Which in turn means, there could also be a string of k1 >k heads, where K1<=n.
