# Drawing uniformly, and sampling with replacement for blue balls and without replacement for red balls

Take a bunch of red and blue balls, and place them in an urn. Let $r$ be the number of red balls, and $b$ be the number of blue balls. If we sample balls uniformly, and if they are red, discard them with probability $p$:

(1) How many blue balls do we sample before discarding all red balls?

(2) How many red balls do we sample before discarding all red balls?

Can we express these values in terms of Harmonic numbers?

Also, is it true that the mean number of times a BLUE ball is selected independent of the total number of BLUE balls? Recall that we are discarding red, not blue balls in the above problem description.

Note: Scouting around it appears that this question is very similar to (Sampling a set with replacement for only a subset of the elements), however, here, we swap the colors and have a probability $p \leq 0$ of discarding a red ball upon sampling it. Do we just divide Brian M. Scott's expectations for the number of, in the context of this question, red ball sampling events, $r$, and blue ball sampling events $b*H(r)$, by $p$? Here, we'd then have that we sample $\frac{r}{p}$ red balls before reaching the end state, and $\frac{b*H(r)}{p}$ blue balls before reaching the end state?

## 1 Answer

I will answer your last question.

Let $f(b,r)$ be the number of times you choose a specific red ball, when there are b blue balls and r red balls inside the urn. $f(b,0)=0$

$f(b,r)=\frac{1}{b+r}(1+f(b,r))+\frac{b-1}{b+r}f(b,r)+p\frac{r}{b+r}f(b,r-1)+(1-p)\frac{r}{b+r}f(b,r)$.

This can result to : $f(b,r)=\frac1{pr}+f(b,r-1)$

As you can see this is independent of $b$.

• Can you use this formula to answer 1st question
• Can you create a similar formula to answer 2nd question?