An urn contains $1$ black and $2$ white balls. One ball is drawn at random and its color is noted. The ball is replaced in the urn, together with an additional ball of its color. There are now four balls int he urn. Again, one ball is drawn at random from the urn, then replaced along with an additional ball of its color. The process continues in this way. Let $B_n$ be the number of black balls in the urn just before the $n$th ball is drawn. I'm supposed to find some conditional expectations here, but I'm not sure how this would work.
Firstly, this is a homework question and I'm just looking for tips on maybe how to look at this in a different way than it's stated or just some different tid-bits of info on how to get started on it.
Best Answer
If you have $b$ black balls and $w$ white balls then the proportion of black balls is $\dfrac{b}{b+w}$.
You then add a black ball with probability $\frac{b}{b+w}$ and a white ball with probability $\frac{w}{b+w}$. So the expected proportion of black balls becomes $$\frac{b}{b+w} \times \frac{b+1}{b+w+1} + \frac{w}{b+w} \times \frac{b}{b+w+1}.$$ Multiply this out and simplify and you will find this is still $\dfrac{b}{b+w}$. So the expectation stays the same as you move forward through the process and so by induction remains the same in future.