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Thread: Events in a row within some arbitrary n.

  1. #1
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    Question Events in a row within some arbitrary n.

    Okay, so say for example, at a football (soccer) game, the chance that someone walking through the turnstiles is an away fan, is 0.1 and 0.9 that they are a supporter of the home team.

    Now I observe 100 people walking through the turnstiles. How would I calculate the probability, of, within this set of 100 people, there being, for example, 7 or more away fans walking through the turnstiles in a row, at least once.

    Thanks and all the best,

    Sean.
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  2. #2
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    Should I have posted this in the University maths help? It's not for University work that's why I put it in here. :S
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  3. #3
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    I've recently started studying probability and statistics myself, so I would like to give this problem a try and help you. But if I were you, I would wait for someone to verify my work.

    If $\displaystyle p(k)$ is the probability of $\displaystyle k$ away fans walking through the turnstiles in a row, then $\displaystyle \sum_{k = 7}^{100} p(k)$ is the probability of $\displaystyle k$ or more away fans walking through the turnstiles in a row.

    To find a formula for $\displaystyle p(k)$, consider a simplified version of your problem with the same probabilities but with 10 observations instead of 100 observations. The following is a possible diagram of the different types of fans walking through the turnstiles

    OOXO

    where Os represent a supporter of the home team and the X represents the k away fans walking through the turnstiles in a row. There are $\displaystyle \displaystyle {10 - k + 1} \choose 1$ possible diagrams. Therefore, the probability of $\displaystyle k$ away fans walking through the turnstiles in a row is given by

    $\displaystyle p(k) = {10 - k + 1 \choose 1} (0.1)^k (0.9)^{10 - k}$

    For your problem, you would use 100 instead of 10.

    You may recognize this probability function as similar to the probability function for a binomial distribution. The difference is the binomial coefficient, which is different because you wanted to know the probability of 7 or more away fans walking through the turnstiles in a row.

    Now, substitute the formula for $\displaystyle p(k)$ into the summation formula at the beginning of this post and evaluate the sum. Can you take it from here?

    Again, I would wait for someone else to verify my work if I were you.
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  4. #4
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    This a very complicated counting question. Stay with me.
    Think of bit-strings, strings of 0s and 1s, of length 100.
    How many of those $\displaystyle 2^{100}$ strings contain no sub-string of seven consecutive 1s?
    Let $\displaystyle \mathcal{S}_n$ be the set bit-strings of length n that contain no sub-string of seven consecutive 1s.
    We want to count $\displaystyle \mathcal{S}_{100}$
    If we can find that number then we how many sequences have 7 or more away fans walking through the turnstiles in a row.
    Here is a start $\displaystyle \| \mathcal{S}_7\|=2^7-1 $. That is the number of bit-strings of length 7 which contain no sub-string of seven consecutive 1s.

    What is $\displaystyle \| \mathcal{S}_8\| $. Look for a pattern!
    What is $\displaystyle \| \mathcal{S}_{100}\|$
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