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 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:

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

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

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    $\begingroup$ Maybe a computer simulation would help convince him. $\endgroup$ – littleO Dec 29 '18 at 10:02
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    $\begingroup$ There is a video of numberphile which came out recently which makes the point about how runs of large length seem improbable to people because they don't seem like a "random" enough permutation. Maybe make him watch that and then you could talk about misguided intuitions. numberphile.com/videos/randomness-is-random $\endgroup$ – Uday Khanna Dec 29 '18 at 10:06
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    $\begingroup$ Here's one thought. If you flip a coin 6 times in a row, your chance of "success" (that is, heads on all six tosses) is $p = 1/64$. If you do 33 independent trials of this experiment (for a total of 198 tosses), the probability of failing all 33 trials is $(1 - p)^{33} \approx 0.6$. So, $0.4$ is clearly a lower bound on your probability of getting 6 heads in a row at least once when flipping a coin 200 times. It's not a very good lower bound, but it might already be larger than what your friend had in mind. $\endgroup$ – littleO Dec 29 '18 at 10:11
  • $\begingroup$ @UdayKhanna i watched it, because of that i said it have high probability $\endgroup$ – Ahmad Dec 29 '18 at 10:13
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    $\begingroup$ @littleO great thinking, i do agree that he thinks the chances are almost zero so $0.4$ is great deal, thanks $\endgroup$ – Ahmad Dec 29 '18 at 10:16

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