Polya’s urn model supposes that an urn initially contains $r$ red and $b$ blue balls. At each stage a ball is randomly selected from the urn and is then returned along with m other balls of the same color. Let $X_k$ be the number of red balls drawn in the first $k$ selections.
(a) Find $\mathbb{E}[X_1]$.
(b) Find $\mathbb{E}[X_2]$.
(c) Find $\mathbb{E}[X_3]$.
(d) Conjecture the value of $\mathbb{E}[X_k]$, and then verify your conjecture by a conditioning argument.
(e) Give an intuitive proof for your conjecture.
Clearly, for the distribution of $X_1$ is Bernoulli with parameter $p$ being equal to $\frac{r}{b + r}$ and hence $$ \mathbb{E} X_1 = \frac{r}{b + r}. $$ In fact, the answer remains the same for $\mathbb{E} X_n, n > 1$. See this link for detailed intuitive explanation.
P.S. I'm also marking this question as a duplicate due to the link provided above.