# Conditional probability + Poisson question

• Dec 9th 2011, 11:36 AM
buenogilabert
Conditional probability + Poisson question
Hi, I have the following problem... I have solved most of it, but I am stuck on this question (Headbang):

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

Thanks a lot for your help! (Happy)

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:

$\displaystyle X$ (buses number $\displaystyle 29$) has Poisson distribution with $\displaystyle μ$ and $\displaystyle Y$ (buses number $\displaystyle 42$) has Poisson with $\displaystyle λ$.

Let $\displaystyle Z=X+Y$, therefore $\displaystyle Z$ will have Poisson distribution with parameter $\displaystyle μ+λ$.

$\displaystyle P(Z=8)=(8^8*e^{-8})/8!=0.1396$"

B:

The solution is supposed to be $\displaystyle 7/64$, but I don't know how to get it.
• Dec 9th 2011, 01:25 PM
emakarov
Re: Conditional probability + Poisson question
You need to find $\displaystyle P(X=6\mid Z=8)$, which is equal to $\displaystyle \frac{P(X=8\cap Z=8)}{P(Z=8)}$. The event $\displaystyle X=6\cap Z=8$ is the same as $\displaystyle X=6\cap Y=2$, and, since $\displaystyle X$ and $\displaystyle Y$ are independent, $\displaystyle P(X=6\cap Y=2)=P(X=6)P(Y=2)$.
• Dec 10th 2011, 11:45 AM
awkward
Re: Conditional probability + Poisson question
Quote:

Originally Posted by emakarov
You need to find $\displaystyle P(X=6\mid Z=8)$, which is equal to $\displaystyle \frac{P(X=8\cap Z=8)}{P(Z=8)}$. The event $\displaystyle X=6\cap Z=8$ is the same as $\displaystyle X=6\cap Y=2$, and, since $\displaystyle X$ and $\displaystyle Y$ are independent, $\displaystyle P(X=6\cap Y=2)=P(X=6)P(Y=2)$.

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.
• Dec 10th 2011, 11:56 AM
emakarov
Re: Conditional probability + Poisson question
Quote:

Originally Posted by awkward
No, X and Y are not independent, since we know X+Y = 8.

I am saying that $\displaystyle P(X=6\cap Y=2)=P(X=6)P(Y=2)$, not that $\displaystyle P(X=6\cap Y=2\mid X+Y=8)$$\displaystyle =P(X=6\mid X+Y=8)P(Y=2\mid X+Y=8)$.
• Dec 10th 2011, 04:06 PM
awkward
Re: Conditional probability + Poisson question
(Bow) OK, at first I didn't see your point, but now I see you are correct.