Challenging Questions

Mathematics – Compulsory Part

Learning Unit: More about probability

Question M0-16-08

We have n bags with the following compositions:

• Bag 1: 2 red balls and 3 white balls
• Bag 2,…,n: 1 red ball and 1 white ball

A ball is drawn randomly from bag 1 and placed in bag 2. Then, a ball is drawn randomly from bag 2 and placed in bag 3. This process continues until a ball is finally drawn randomly from bag n. Let \(R_n\) and \(W_n\) be the events of drawing a red ball from bag n and of drawing a white ball from bag n, respectively.

1. Calculate \(P(R_1)\) and \(P(R_2)\).
2. Calculate \(P(W_2)\).
3. Show that \(P(R_2)-\frac{1}{2}=\frac{1}{3}[P(R_1)-\frac{1}{2}]\) and \(P(R_3)-\frac{1}{2}=\frac{1}{3}[P(R_2)-\frac{1}{2}]\).
4. Hence, suppose that \(P(R_n)-\frac{1}{2}=\frac{1}{3}[P(R_{n-1})-\frac{1}{2}]\). Find \(P(R_n)\) in terms of \(n\).

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