Suppose I roll a die and then draw that many cards from a deck without replacement. What is the expected value of the number of heart cards given that I roll a 3?
So I had this idea of doing:
X=number obtained on the die
Y=number of hearts
$E(Y|X=3)= \sum_{y=1}^{3}yP_{(y|x)}(Y|X=3)=6$
Since from my understanding $$ P_{(y|x)}(Y|X=3)=\frac{P(y\bigcap x=3)}{P(x=3)}=\frac{P( x=3)}{P(x=3)}=1$$
I have no way to check that if anyone spots an error or would like to find me a better way of doing it that would be awesome.
Guide:
Given that you are drawing $3$ cards,
Probability of getting no heart is $\frac{39 \cdot 38 \cdot 37}{52 \cdot 51 \cdot 50}$
Probability of getting a heart is $3 \cdot \frac{39 \cdot 38\cdot 13}{52 \cdot 51 \cdot 50} $
Probability of getting $3$ hearts is $\frac{13\cdot 12 \cdot 11}{52\cdot 51 \cdot50}$
Can you compute the expected value given these info?