How do I find the joint pdf of the joint distribution X, Y, Z

Question

Six cards are to be drawn from an ordinary deck of cards. $X$ is the number of low ($1$-$5$) black suited cards drawn $Y$ is the number of high ($6$-$10$) red suited cards drawn $Z$ is the number of face cards drawn. What is the joint pdf of $X, Y, Z$?

I know that the number of ways to choose for $X$ is $10Cx$ The number of ways to choose for $Y$ is $10Cy$ And the number of ways to choose for $Z$ is $12Cz$.

I know I have to multiply all $(10Cx)(10Cy)(12Cz)$ then divide it by $(52C6)$.

But it seems like I'm missing out something else...

You are missing that it is possible to select some cards that are not low black, high red, or faces. — Graham Kemp, Nov 30 '17 at 12:39

score 1 · Answer 1 · answered Nov 30 '17 at 12:08

1

$P(X=x,Y=y,Z=z) = \dfrac{{10\choose x}{10\choose y}{12\choose z}{20\choose (6-x-y-z)}}{{52\choose 6}}$

This is the joint pdf of X,Y,Z

answered Nov 30 '17 at 12:08

Satish Ramanathan

11,984

Why is there a 20C(6-x-y-z)? – Yuumi Nov 30 '17 at 12:25
That term shall not be there if you are sure that x+y+z = 6. But the problem does not guarantee that. – Satish Ramanathan Nov 30 '17 at 12:28
@Yuumi, ways to select the rest of the hand from cards that are not low black, high red, or faces. – Graham Kemp Nov 30 '17 at 12:37

How do I find the joint pdf of the joint distribution X, Y, Z

1 Answers1