# Probability question; Conditional probability and poisson distribution

by TaliskerBA
Tags: conditional, distribution, poisson, probability
 P: 26 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...
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