Question

A bag contains 3 red and 2 green balls. AN expreiment consists of drawing a ball from this bag and replacing it with 1 more ball of same colour in this bag. This experiment in done thrice. Now one more ball is drawn from the bag, Find the chance that it is red

Answer 1

3/5

Answer 2

The problem describes a process known as Polya's Urn scheme.

Initial State:

• Number of red balls ($R_0$) = 3
• Number of green balls ($G_0$) = 2
• Total number of balls ($N_0$) = $R_0 + G_0 = 3 + 2 = 5$

The Experiment:

An experiment consists of drawing a ball, replacing it, and then adding 1 more ball of the same colour to the bag. This experiment is performed thrice.

The Question:

After these three experiments, one more ball is drawn from the bag. We need to find the probability that this ball is red.

Polya's Urn Property:

A key property of Polya's Urn scheme is that the probability of drawing a specific color ball on any given draw is the same as the initial probability of drawing that color ball.

Let $P(R_k)$ be the probability of drawing a red ball on the $k$-th draw.

For a Polya's Urn scheme: $P(R_k) = \frac{R_0}{N_0}$ for any $k \ge 1$.

In this problem, $R_0 = 3$ and $N_0 = 5$.

We need to find the probability of drawing a red ball on the fourth draw, i.e., $P(R_4)$.

According to the property of Polya's Urn:

$P(R_4) = P(R_1) = \frac{\text{Initial number of red balls}}{\text{Initial total number of balls}}$

$P(R_4) = \frac{3}{5}$

Thus, the probability of drawing a red ball on the 4th draw is 3/5.

Explanation:

In a Polya's Urn scheme, if you start with $R_0$ red balls and $G_0$ green balls, and after drawing a ball, you replace it and add $c$ more balls of the same color, then the probability of drawing a red ball on any subsequent draw ($k$-th draw) remains constant and is equal to the initial probability of drawing a red ball.

Initial probability of drawing a red ball = (Number of red balls) / (Total number of balls) = 3 / (3+2) = 3/5.

Since the experiment is performed thrice, and then one more ball is drawn (which is the fourth draw), the probability of this fourth ball being red is still 3/5 according to Polya's Urn property.