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If you were to flip a coin 150 times, what is the probability that it would land tails 7 times in a row? How about 6 times in a row? Is there some forumula that can calculate this probability?

  • 0
    The calculation is not too difficult but somewhat involved. It is much easier to calculate the probability that the coin lands tails at most 6 (resp. at most 5) times in a row. This probability is described by a linear recurrence which has a closed formula depending on the roots of its characteristic polynomial, and evaluating it at n = 150 gives the answer.2010-09-14
  • 0
    Thanks, I figure the odds would be 1 in 128 if there were just 7 coin flips... but I'm stuck as to how to calculate for n = 1502010-09-14
  • 5
    To see why the probability is much larger than 1/128, break the 150 coin flips into 21 groups of 7 (plus 3 left over) and ask what the chance is that *none* of those groups has seven tails. Answer: (1 - 1/128)^21 = about 0.85. Its complement, 0.15 = 1-0.85, underestimates the solution because the seven in a row could span two groups. An overestimate is obtained by looking at the 144 overlapping groups of 7 flips: 1 - (1-1/128)^144 = 0.68; it's an overestimate because the groups are correlated. The truth lies somewhere in between, as the answers below more rigorously attest.2010-09-14
  • 0
    A longish article about such probabilities can be found there: http://gato-docs.its.txstate.edu/mathworks/DistributionOfLongestRun.pdf2010-09-15
  • 0
    also refer to [this post](http://math.stackexchange.com/questions/2045496)2017-01-07

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