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
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.
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 is the probability of away fans walking through the turnstiles in a row, then is the probability of or more away fans walking through the turnstiles in a row.
To find a formula for , 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 possible diagrams. Therefore, the probability of away fans walking through the turnstiles in a row is given by
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 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.
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 strings contain no sub-string of seven consecutive 1’s?
Let be the set bit-strings of length n that contain no sub-string of seven consecutive 1’s.
We want to count
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 . That is the number of bit-strings of length 7 which contain no sub-string of seven consecutive 1’s.
What is . Look for a pattern!
What is