There are $n>1$ workers in a firm, share $\phi$ of them are ''bad'' and the remaining $1-\phi$ are ''good''. During a given time period, a bad worker quits with probability $\beta$ and a good worker quits with probability $\alpha$. The quit events occur independently both across and within worker types. Let $1\leq k\leq n$ workers actually quit. What is the share of bad workers among those $k$ leavers?
Here's where I got so far. For $k=1$, the share is the probability that this one leaver is a bad worker, and is equal to $\frac{\phi \beta}{\phi \beta+(1-\phi)\alpha}$ according to Bayes' Theorem. For $k=n$, when all workers quit, the share of bad workers among them is clearly $\phi$. But what about intermediate cases?