Say that there is an urn with balls of different colors. $P(R)$ and $P(B)$ are the probabilities of drawing red or blue balls. These do not add up to one.
Say I have $N$ draws (with putting back the ball after each draw). I want to know the probability of drawing $X$ red balls, and $1$ blue ball, where $X+1 \leq N$.
I am able to do this when I use (small) natural numbers for $X$ and $N$ and just write out all the combinations.
I could use the binomial distribution if I only wanted to know the probability of drawing $X$ red balls given $N$ draws. My problem is the joint probability of both events.