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.
Can someone explain to me the solution step by step? I am completely lost... I mean I am guessing X and Y are binomial since we have "large population", But I am lost from the line in solution starting "From conditioning on whether or not..."
$$E(Y)=E(Y|A)P(A)+E(Y|A^c)P(A^c)$$ Law_of_total_expectation
$A=\{X\geq 1\}$ so $A^c=\{X < 1\} =\{X=0\}$
$$E(Y)=E(Y|X\geq 1)P(X\geq 1)+E(Y|X=0)P(X=0)$$
$$E(Y)-E(Y|X=0)P(X=0)=E(Y|X\geq 1)P(X\geq 1)$$
$$\frac{E(Y)-E(Y|X=0)P(X=0)}{P(X\geq 1)}=E(Y|X\geq 1)$$