Probability of getting red ball at ith step

All orderings of the balls are equally likely. So the probability that the $i$-th ball is red is the same as the probability that the first ball is red.

Think of the $n=r+w$ balls as people. The probability that Charlie is the $i$-th person chosen is the same as anyone else's. So the probability that Charlie is the $i$-th person chosen is $\frac{1}{n}$.