Events in a row within some arbitrary n.

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June 9th 2011, 08:28 PM
sean.1986

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.
June 10th 2011, 02:37 AM
sean.1986

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
June 10th 2011, 03:45 PM
NOX Andrew

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

To find a formula for $p(k)$ , 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 $\displaystyle {10 - k + 1} \choose 1$ possible diagrams. Therefore, the probability of $k$ away fans walking through the turnstiles in a row is given by

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

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 $p(k)$ 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.
June 10th 2011, 05:12 PM
Plato

This a very complicated counting question. Stay with me.
Think of bit-strings, strings of 0’s and 1’s, of length 100.
How many of those $2^{100}$ strings contain no sub-string of seven consecutive 1’s?
Let $\mathcal{S}_n$ be the set bit-strings of length n that contain no sub-string of seven consecutive 1’s.
We want to count $\mathcal{S}_{100}$
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 $\| \mathcal{S}_7\|=2^7-1$ . That is the number of bit-strings of length 7 which contain no sub-string of seven consecutive 1’s.

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