Say the $N$ people in Springfield either have 2 corgis (with probability $p_1$), a corgi and a shizu (with probability $p_2$), or 2 shizus (with probability $p_3$). Let $J$ be the number of people with 2 corgis, $K$ be the number of people with a corgi and a shizu, and $L$ be the number of people with 2 shizus. What is the distribution of the number of corgis ($C$) out of a total of $2N$ dogs in Springfield?
$C = 2J + K$. The distribution of people with 2 corgis is $Binomial(N, p_1)$, and the distribution of people with a corgi and a shizu is $Binomial(N, p_2)$. The sum of binomial random variables is also binomial, so I have a hunch that $C$ should be binomial.
However, how do I find the probability associated with $C$?