Can't get a grasp of this probability

[kioria]

This is a simple example in a textbook, I haven't done probability since High school and probability being all too confusing... I cannot seem to overcome this problem. Here's the problem:

a) Assume k car accidents occurred in n days. Assume that accidents are equally likely on any day. Let A = event that one accident occurred each day. What is P(A)?

The solution is given as below:
Solution: $$P(A) = \frac{n(n-1)...(n-k+1)}_{n^k}$$


Can someone explain this solution or the process of obtaining this solution in plain english? Thanks.

 

[EnumaElish]

$$P(A) = \frac {\text{number of ways event A could have happened}}{\text{total number of ways k accidents could have happened in n days}}$$

$$={\text{number of days the first accident could have happened}\times...$$
$$...\times\text{ number of days the k'th accident could have happened, which otherwise would have been accident-free}}$$
$$\left/{\text{(I need to think a little more about the denominator here)}}\right.$$

 

[kioria]

Perhaps,

P(A) = the probability of accident happening on first day AND the probability of accident happening on second day AND ... AND the probability of accident happening on the last day.

This turns out to be:
$$P(A) = \frac{n(n-1)...(n-k+1)}_{n^k}$$, since only 1 accident per days needs to happen for all n days, AND is a likely clause... and I think its right. Unless someone corrects me!

 

[EnumaElish]

Yes, AND is right. I've been thinking about the denominator, though... Is it the number of subsets with k elements each, out of a total of n elements?

 

[EnumaElish]

the probability of accident happening on first day AND the probability of accident happening on second day AND ... AND the probability of accident happening on the last day.

It's more like, the number of days that the 1st accident can happen all by itself: since it's the 1st accident, it could happen any day, so the 1st accident has n "choices." Then, the 2nd accident has n-1 "choices" because one of the days has been "reserved" by the 1st accident, and so on.

P.S. In this post, the order 1st, 2nd, ... does not necessarily refer to temporal priority. "1st accident" does not necessarily mean "earliest accident." It just means "the first accident being looked at." And it could have happened in any of the n days, so in principle it could have happened on Wednesday whereas the 2nd accident being looked at could have happened the day before (Tuesday).

 

[kioria]

I believe the question says:

$$N = \{k_1, k_2, ... k_n\}$$ where n is an Integer for n days. Let k denote $$k_1 + k_2 + ... + k_n$$.

I am puzzled as to why $$\frac{(n - k)}_{n}$$ is omitted. I can't seem to picture the relationship between the final result and k.

 

[kioria]

It's more like, the number of days that the 1st accident can happen all by itself: since it's the 1st accident, it could happen any day, so the 1st accident has n "choices." Then, the 2nd accident has n-1 "choices" because one of the days has been "reserved" by the 1st accident, and so on.

P.S. In this post, the order 1st, 2nd, ... does not necessarily refer to temporal priority. "1st accident" does not necessarily mean "earliest accident." It just means "the first accident being looked at." And it could have happened in any of the n days, so in principle it could have happened on Wednesday whereas the 2nd accident being looked at could have happened the day before (Tuesday).

I see what you mean there...

[EDIT] But shouldn't the choice be chosen from total number of accidents that have happened over the period of n days?

 

[EnumaElish]

You are right!

$$P(A) = \frac{n}{n}\times...\times\frac{n-k+1}n$$

Of course!

The reason why n-1 was left out is, if you have 7 days (Mon-Sun) and 3 accidents, then the 1st accident might happen on Wed., and second on Tue. Number of days left for the 3rd accident = 5 = 7 - 2 = 7 - (3 - 1) = 7 - 3 + 1. That's because the last accident will have k - 1 days previously "reserved" by k - 1 accidents before it has a chance to "decide" which day it's going to happen.

 

[EnumaElish]

I see what you mean there...

[EDIT] But shouldn't the choice be chosen from total number of accidents that have happened over the period of n days?

Exactly. So if you had 3 accidents (Tue, Wed, Thu) in 7 days (Mon-Sun) then you might say, let me see on how many days the accident that happened on Tue could have happened? The answer is 7 days. Next, having reserved the day on which the 1st accident COULD HAVE happened, what is the number of days that the 2nd accident could have happened? Since I know that the "first" accident happened on ONE DAY, I have 6 days left for the 2nd one.

 

[kioria]

Ahhh I see it now. So k acts to restrict the cardinality of the set S in relation to the final answer. Since you start off with $$\frac{n}_{n}$$ which is probability of an accident happening on any of the n days, as days go by you can only have accidents if k is sufficiently big enough for n. Otherwise if ($$k >= n$$) then the solution would be simple $$\frac{n!}_{n^k}$$.

Thanks for that.