Probability question; Conditional probability and poisson distribution

TaliskerBA

1. The problem statement, all variables and given/known data

A radioactive source emits particles according to a Poisson process, at an average rate of λ per unit time. Each particle emitted has probability p of being detected by an instrument, independently of other particles. Let X be the number of particlese emitted in a given time interval of length T , and Y the number of those particles that are detected. As usual, let μ = λT and q = 1 − p.
(i) What is the conditional probability p(Y = r|X = k)?


2. Relevant equations

I know that I want p(Y=r|X = k) = p((Y=r) ∩ (X=k)) / p(X=k)

I know from poisson distribution that p(X=k) = ((μ^k)/k!)*e^(−μ)


3. The attempt at a solution

I don't understand how I can work out what p((Y=r) ∩ (X=k)) equals but this is my attempted solution:

Since there are k particles emitted and we want to know the probability that r of them have been detected then using binomial distribution:

p((Y=r) ∩ (X=k)) = [k choose r](p^r)(q^(k-r))

I know this is wrong but I can't quite work out how to tie it all together...

fzero

You wrote down a formula assuming that k particles have been emitted. Therefore this is not the joint probability, but it is still relevant to the conditional probability that you want to compute...