Probability of 7 heads in a row over 1000 flips, 2 seperate times?
P(heads) = 0.5, and p(7heads in a row in 1000 flips) = 0.98. (I got this value from some calculator online, not actually sure if this is correct or not, as when I tried using excel with input =1-BINOM.DIST(0,994,0.5^7,FALSE), I got different answer, but it's been a while since i learnt probability)
But what would be the probability of this event (7heads in a row) occuring not once, but 2 seperate times over the course of 1000 flips?
Thanks in advance.
2 Replies
I was working on streak stuff recently and I gave the Villarino paper (linked upthread) another read. The author derived equations 4 and 5 by manipulating a generating function and then called it a day, with no regard for the combinatorial meaning of the final result. After staring at the formula for a while, I was finally able to understand it. Then I realized the formula could be generalized to multiple streaks with a simple tweak. This might have been worthy of writing a paper to get published, which I was ready to try, only to learn that it's already been done:
The Muselli paper is from 1996, predating Villarino's by 12 years while offering the more general formula along with a combinatorial explanation. I found it with a quick search on Google Scholar, contrary to Villarino's claim that "the explicit formula...is amazingly hard to find in the literature."
The formula is Equation 13, but that's the PMF. Since we want the chance of at least x streaks, replace the C(m,x) term with C(m–1,x–1).
Nothing to add to the already well explained calculation, but this is one area where for most people our instincts about what is random fail us. I had a stats professor once who demonstrated this with an assignment. The assignment was simple (if tedious) - flip a coin 10000 times and record the results.
There were people in the class who tried to cheat and just write down a series of “random” heads and tails. Those people all got busted. How? Nobody doing it that way would think to make long streaks of heads or tails. If doing it honestly, though, steaks of six or more are almost certain, and streaks of even 11 or 12 are quite probable. Most people domt look at a coin flipping heads 9 or 10 times in a row and think it’s truly random, but given enough flips, that’s exactly what you would expect to see.
