Question

Suppose that you get into a car accident at an average rate of about one accident every 3 years. In the next 21 years, you would expect to have about 7 accidents. if you actually had 5 accidents in that time, can you say that you're a better driver?

Explain briefly using Poisson distribution or other methods.

Answer 1

You could say so, but people some people are better drivers/have more experience, etc.

Explanation:

You could think of it like that, but for example a younger 18 year old is more likely to have crashes, more than a 39 year old. So therefore that person is bringing up the average crashes.

So maybe if for example you are 21, look at the average time for a crash between 18-22 year olds or similar, or you could do a questionnaire and compare yourself to others, then you can see if you are really a better driver.

Answer 2

Answer considering Poisson distribution...

Explanation:

This problem is heavily influenced by factors we simply dont know, like Brandon mentioned, a like age etc... but we can attempt to model this situation with a poisson distribution

We know Poisson: `X ~ Po(lamda)`

In this case `X` is the distribution for a 3 year interval...

So hence your rate, `lamda = 1`

Hence we can recal our properties about poisson distribution...

if `X ~ Po(lamda_0)` for 1 unit of time

then `X ~ Po(k lamda_0)` for `k` units of time

So hecne let `Y` be our variable for 21 years:

`Y ~ Po(1* 21/3 ) -= Y~ Po(7)`

Now recal `P(X=x) = (e^(-lamda) * lamda^x )/(x!)`

So hecne our probability of getting in 7 accidents:

`P(Y = 7 ) = (e^(-7) * 7^7 )/(7! ) approx 0.149`

5 accidents

`P(Y = 5 ) = (e^(-7) * 7^5 )/(5! ) approx 0.12772`

So actually the probability of being in 7 accidents over that time is relitivly small , so from this statement we can say that the driving is good, but not as much as first considered... But getting in 5, is even less likely in calculations...