# What is the joint probability mass function?

A pond has $r$ red fish, $b$ blue fish, and $g$ green fish. Let $R$ be the number of red fish, $B$ be the number of blue fish, and $G$ be the number of green fish in a random sample size of $N$. What is the joint probability mass function of $R$, $B$, and $G$?

I understand that $G$ should be hypergeometrically distributed since only a certain number of the fish are green, but how do I construct a joint distribution?

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## 1 Answer

There are $\binom{r+g+b}N$ different samples of size $N$. The number of samples of size $N$ with $R$, $G$ and $B$ fish of the various colours is $\binom rR\binom gG\binom bB$ if $R+G+B=N$ and $0$ otherwise. Thus the desired probability is

$$p_N(R,G,B)=\begin{cases}\binom rR\binom gG\binom bB\binom{r+g+b}N^{-1}&R+G+B=N\;,\\\\\\0&\text{otherwise.}\end{cases}$$

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