Thread: Conditional probability + Poisson question

Hi, I have the following problem... I have solved most of it, but I am stuck on this question :

"B) Given that eight buses arrive, what is the probability that six of them are number 29's?"

Thanks a lot for your help!

The problem and what I have done so far is:

"Number 29 and number 42 buses arrive at a bus stop according to independent Poisson processes each at a rate of four per hour.

A) What is the probability that eight buses arrive between 10 and 11am?
B) Given that eight buses arrive, what is the probability that six of them are number 29's?"

My answer to A:

(buses number ) has Poisson distribution with and (buses number ) has Poisson with .

Let , therefore will have Poisson distribution with parameter .

"

B:

The solution is supposed to be , but I don't know how to get it.

You need to find , which is equal to . The event is the same as , and, since and are independent, .

Originally Posted by emakarov

You need to find , which is equal to . The event is the same as , and, since and are independent, .

No, X and Y are not independent, since we know X+Y = 8.

Given that X+Y=8, X has a Binomial distribution with p = 1/2 and n = 8.

Originally Posted by awkward

No, X and Y are not independent, since we know X+Y = 8.

I am saying that , not that .

OK, at first I didn't see your point, but now I see you are correct.