Conditional expectation.Tthe expected number of patients with advanced stage cancer, given at least one patient has early stage cancer In a large population of patients, 20% have early stage cancer, 10% have advanced stage cancer, and the other 70% do not have cancer. Six patients from this population are randomly selected. Calculate the expected number of selected patients with advanced stage cancer, given that at least one of the selected patients has early stage cancer. This is what I have so far: Let $X$ = number of patients that has early stage cancer Let $Y$ = number of patients that has advanced stage cancer And I tried to use the definition of conditional expectation values $E(Y|X \geq1) = \sum^{y=5}_{y=0} yP(Y=y|X\geq1)$ But how do I calculate $P(Y=y$ $and$ $X \geq1$)? Or is there other better ways to solve this problem? Thanks for any help

You could do this as double sums, since you can calculate $P(X=x \text{ and }Y =y)$.

$$E(Y|X \geq 1) = \dfrac{\displaystyle \sum^{5}_{y=0} \sum^{6-y}_{x=1} y\,P(X=x \text{ and }Y =y) }{\displaystyle \sum^{5}_{y=0} \sum^{6-y}_{x=1} \,\,\,P(X=x \text{ and }Y =y) }$$ though you may prefer the equivalent $$E(Y|X \geq 1) = \dfrac{\displaystyle \sum^{6}_{x=1} \sum^{6-x}_{y=0} y\,P(X=x \text{ and }Y =y) }{\displaystyle \sum^{6}_{x=1} P(X=x ) }$$

and in both cases the denominator is $\displaystyle P(X \geq 1)$.