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Math Help - P(H heads in a row) given N coin flips

  1. #1
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    P(H heads in a row) given N coin flips

    Hi,

    I recently thought of the following problem and my basic grasp of probability and combinatorics is not enough to handle it:

    You flip a coin N times. What is the probability P(N,H) that you get at least one sequence of at least H heads in a row?

    Since this question deals with the number of times a result will happen, I tried to use a Poisson Distribution, but I'm having trouble separating the problem into mutually exclusive events.

    I wrote a program to do a simulation and found that P(20,7) ≈ 0.058, so please make sure your answer fits with this result.

    Thank you for your help!
    Last edited by lambda; October 3rd 2009 at 03:15 PM. Reason: added what I've tried
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  2. #2
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    Quote Originally Posted by lambda View Post
    Hi,

    I recently thought of the following problem and my basic grasp of probability and combinatorics is not enough to handle it:

    You flip a coin N times. What is the probability P(N,H) that you get at least one sequence of at least H heads in a row?

    Since this question deals with the number of times a result will happen, I tried to use a Poisson Distribution, but I'm having trouble separating the problem into mutually exclusive events.

    I wrote a program to do a simulation and found that P(20,7) ≈ 0.058, so please make sure your answer fits with this result.

    Thank you for your help!
    I would do it this way: treat the "H heads in a row" as a single object and call it "R" for "row". With N flips and "H heads in a row", there are N- H other flips. How many different ways are there to order 1 object and N- H other objects?
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  3. #3
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    Problem - counts certain cases twice

    Quote Originally Posted by HallsofIvy View Post
    I would do it this way: treat the "H heads in a row" as a single object and call it "R" for "row". With N flips and "H heads in a row", there are N- H other flips. How many different ways are there to order 1 object and N- H other objects?
    Thank you for your reply.

    I tried this approach and got P(n,h) = \frac{acceptable\_series}{total\_possible\_series} = \frac{2^{n-h}(n-h + 1)}{2^n} = \frac{n-h + 1}{2^h} but kept getting answers that were roughly 2x too large.

    I've realized that this approach counts certain answers twice. For example, for n=7 and h=3, the series HHHHHHH is counted 5 times, as RHHHH, HRHHH, HHRHH, HHHRH, and HHHHR. Do you have any tips on how to fix this?
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