Can someone help me solve this question please??
Pretend you are writing traffic accident software and want to categorize accidents by the day of the week on which they occur. Pretend there are n accident reports to categorize.
- What is the size of the sample space? That is,in how many ways can the n accident reports be distributed over 7 days?
- In how many ways can all n accidents occur on one single day?
- In how many ways can all n accidents occur on only two days?
- Let’s looks at the other end:In how many ways can all n accidents occur on seven, and no less, days.
1.) If n = 1, it is clear that the sample space is 7. If n = 2, we know the first accident goes one of 7 places, and the second does too. Then, we have 7² ways.
Following this reasoning, you should be able to find the simple answer to this one for general n.
2.) Given that they all occur on one day, and that there are 7 days, there are 7 ways that this can happen.
3.) First, we pick two days.
We can have 1 accident on the first, n - 1 on the second, and then 2 on the first, n - 2 on the second, all the way to n - 1 on the first, and 1 on the second. Thus there are (n-1) ways this can happen, given two days.
There are C(7,2) possibilities of two days, giving us C(7,2)(n-1) as a final result.
4.) There is only one choice of seven days, so we can ignore that. We also have to assume that there are at least 7 accidents that week.
If there are exactly 7 accidents, this can happen only one way, since accidents are categorized by the day of week only.
If n = 8, then an additional accident can be placed on any day, giving 7 ways.
If n = 9, there are two additional accidents that can both go anywhere, giving 7² ways.
Following this reasoning, it suggests that the number of ways that n accidents can fall on precisely 7 days is in fact 7^(n-7).