3.2 Exercise 22

Polya's urn scheme for a contagious disease. An urn contains initially b black balls and r red balls ($r + b = n$). A ball is drawn on an equally likely basis from among those in the urn, then replaced along with c additional balls of the same color. The process is repeated. There are n balls on the first choice, $n + c$ balls on the second choice, etc. Let B_k be the event of a black ball on the kth draw and R_k be the event of a red ball on the kth draw. Determine

1. $P \left(\right. B_{2} \left|\right. R_{1} \left.\right)$
2. $P \left(B_{1} B_{2}\right)$
3. $P \left(R_{2}\right)$
4. $P \left(\right. B_{1} \left|\right. R_{2} \left.\right)$.