Question
An urn initially contains $r$ red and $b$ blue balls. At each stage, a ball is randomly selected and returned along with $m$ other balls of the same colour. Let $X_k$ be the number of red balls drawn in the first $k$ draws. Conjecture the value of $\mathbb{E}(X_k)$ and verify your conjecture using a conditioning argument.
Hints
For $1 \leq i \leq k$, define $$Y_i = \begin{cases} 1 & \quad \mathrm{if\ draw\ } i\ \mathrm{from\ the\ urn\ is\ red}\\ 0 & \quad \mathrm{if\ draw\ } i\ \mathrm{from\ the\ urn\ is\ blue} \end{cases}$$ and evaluate $\mathbb{E}[Y_3 \mid X_2]$.
My working
Conjecture: $\mathbb{E}(X_k) = \dfrac {kr} {r + b}$.
Let $R_k$ and $B_k$ be the events that a red or blue ball was drawn in the $k^{th}$ draw respectively.
For $k = 2$, we have
$\begin{aligned} \mathbb{P}(R_2) & = \mathbb{P}(R_2 \mid R_1)\mathbb{P}(R_1) + \mathbb{P}(R_2 \mid B_1)\mathbb{P}(B_1)\\[1 mm] & = \frac {(r + b)\left(\frac r {r + b}\right) + m} {r + b + m}\left(\frac r {r + b}\right) + \frac {(r + b)\left(\frac r {r + b}\right)} {r + b + m}\left(1 - \frac r {r + b}\right)\\[1 mm] & = \frac m {r + b + m}\left(\frac r {r + b}\right) + \frac {(r + b)\left(\frac r {r + b}\right)} {r + b + m}\\[1 mm] & = \frac r {r + b}, \end{aligned}$
so the base case is true.
Now, suppose the conjecture is true for $k = n$ and for $k = n + 1$, we have
$\begin{aligned} \mathbb{P}(R_{n + 1}) & = \mathbb{P}(R_{n + 1} \mid R_n)\mathbb{P}(R_n) + \mathbb{P}(R_{n + 1} \mid B_n)\mathbb{P}(B_n)\\[1 mm] & = \frac {[r + b + (n - 1)m]\left(\frac r {r + b}\right) + m} {r + b + nm} \left(\frac r {r + b}\right) + \frac {[r + b + (n - 1)m]\left(\frac r {r + b}\right)} {r + b + nm} \left(1 - \frac r {r + b}\right)\\[1 mm] & = \frac m {r + b + nm} \left(\frac r {r + b}\right) + \frac {[r + b + (n - 1)m]\left(\frac r {r + b}\right)} {r + b + nm}\\[1 mm] & = \frac r {r + b} \end{aligned}$
Thus, by induction, we can see that the probability of drawing a red ball at every stage is constant at $\dfrac r {r + b}$ and independent of drawing any ball at any other stage, so $\mathbb{E}(X_k) = \dfrac {kr} {r + b}$.
I know that there are already quite a few proofs out there regarding Polya's urn model and although I believe my proof is valid, I am still posting this as I am not exactly sure how to make use of the hints given. Any intuitive suggestions would be greatly appreciated :)
