$n$ vehicles are stopped at random, the probability that a driver who is stopped is a beginner is $p$ while the probability that a driver who is stopped is a professional is $q$. There are drivers that are neither beginners nor professionals. $X$ is the random variable representing the number of beginners stopped while $Y$ is the number of professional drivers stopped.
- If $X_k$ is the random variable of the number of beginners stopped knowing that exactly $k$ professionals were also stopped, what is is the conditional probability $\mathrm {Pr}(X=i|Y=k)?$
- What is the probability $\mathrm {Pr}(X=i, Y=j)$?
- If $Y_k$ is the variable defined by $\cfrac Y{X+Y}$ with $X+Y = k$, what is the probability distribution and expected value of $Y_k$?