Probability question; Conditional probability and poisson distribution
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...
Re: Probability question; Conditional probability and poisson distribution
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