Probability of at least one streak of 5 heads in 100 tosses of a fair coin What is the probability of at least one streak of 5 heads in 100 tosses of a fair coin? ($n=100,p=0.5,k=5$.)
Additional question: what is the general formula for any $n, p, k$ (and proof)? 

One can model this as a Markov chain. This will have states labelled $0$, $1$, $2$, $3$, $4$ and $5$. State $0$ is initial; you start there, and state $5$ is absorbing; if you reach it you remain there. At state $k$ ($0\le k\le 4$), tossing a head takes you to state $k+1$ and tossing a tail to state $0$. So the state basically keeps track of the current run of heads. So the question is, what is the probability you reach state $5$
after $100$ tosses. This can be solved by standard Markov chain methods,
for instance using the transition matrix.
