Probability of getting 6 heads in a row from 200 flips and intuition about this high value A few days ago I had an argument with a friend about this question :

What is the probability of getting 6 heads in a row from 200 flips ?

I argued it is high probability (significantly bigger than a half) while he argued it is low probability.
When I tried to give exact formula I failed, so we checked the web were the answer was about 84%, yet he is still not convinced so from this I have two questions:

*

*What is the exact formula for $k$ Heads in a row (consecutive) out of $n$ coin flips?


*(Not a mathematical) How to convince my friend that 6 in a row have high probability? Meaning what is the intuition behind the question ?

Here is how to calculate the exact answer.  Consider a Markov chain $X_0,X_1,\ldots,X_{200}$, taking integer values in the range $0\le X_n\le 6$, with
transition matrix (with row and column indices in the range $0\le i,j\le6$) $$M=\pmatrix{\frac12&\frac12&0&0&0&0&0&\\
\frac12&0&\frac12&0&0&0&0&\\
\frac12&0&0&\frac12&0&0&0&\\
\frac12&0&0&0&\frac12&0&0&\\
\frac12&0&0&0&0&\frac12&0\\
\frac12&0&0&0&0&0&\frac12\\
0&0&0&0&0&0&1}$$
Here the idea is that $X_n$ represents the number of consecutive heads ending at  flip $n$ (with the conventional courtesy value $X_0=0$) so that the flip sequence HTHH would cause $X_0=0$, $X_1=1$, $X_2=0$, $X_3=1$, $X_4=2$, and so on.  Except the value $X_n=6$ means something different: either $X_{n-1}=6$ or the $n$-th flip was H and $X_{n-1}=5$.  The chain is started with the  value $X_0=0$; what is sought is the probability that $X_{200}=6$.  This is the $(0,6)$-th entry in the matrix $M^{200}$.  When I do these calculations I get a value very close to $.8$.
Here is a way to visualize this chain.  There is a small spider that aspires to climb to the top of a $6$ segment pipe.  It starts at the bottom, and does the following $200$ times:

*

*if it is at the top, it stays at the top,

*otherwise it flips a coin and drops all the way to the bottom if the coin shows T, and climbs up more more segment if the coin shows H.

