To find the maximum of repeated balls extractions' probability

Question

"In a urn there are some white balls and black balls. Three balls are extracted from the box, reintroducing them back after their extraction. Find the maximum of the probability that exactly two balls out of the three are white."
My unsuccessfull idea was to maximize the function $$p(B)=\frac{C'_{B,2}C'_{k-B,1}}{C'_{k,3}}=-3\frac{B^3-(k-1)B^2+kB}{k(k+1)(k+2)}$$ and then maximize the maximum of $p(B)$ but I found that this last step can't help at all because a maximum for the $\max p(B)$ doesn't even exist… Have you other and better ideas?

Answer 1

Let $p$ be the probability that a drawn ball is white. Since the balls are drawn with replacement, the probability that two out of three are white is $$f(p)=3p^2(1-p)$$

The derivative is $0$ when $p=0$ or $p=\frac23.$ The first solution is obviously extraneous, so the answer is $p=\frac23$. This is achieved when there are twice as many white balls as black balls.