# Probability of combinations with replacement for picking majority of some color

Say that I have two set of balls, $r$ red and $b$ blue. I am interested to find a function that tells me what is the probability of picking $red>blue$ balls when picking a total of $s$ out of $n=r+b$ balls?

Exact probability: The probability of picking a single red ball is $\frac{r}{n}$, the probability of picking a two red balls is $\frac{r}{n} \times \frac{r-1}{n-1}$ and more generically the probability of picking all red balls when picking up $s$ balls is: $\frac{C^r_s}{C^n_s}$.

Majority probability: How do I calculate $P[s\text{ balls selected and there are more red than blue balls}]$?

I believe this is the same as saying that $P[\text{at least s/2 balls are red}]$