Q. 3.23 - A First Course in Probability

Question

Urn $\mathrm{I}$ contains $2$ white and $4$ red balls, whereas urn $\mathrm{\Pi }$ contains $1$ white and $1$ red ball. A ball is randomly chosen from urn $\mathrm{I}$ and put into urn II, and a ball is then randomly selected from urn $\mathrm{\Pi }$. What is

(a) the probability that the ball selected from urn $\mathrm{\Pi }$ is white?

(b) the conditional probability that the transferred ball was white given that a white ball is selected from urn $\mathrm{\Pi }$?

Step-by-Step Solution

Answer

a)  The probability that the ball selected White is $\frac{4}{9}$

b)  The probability that the transferred White ball was $\frac{1}{2}$

Step 1: Set an occurrence

When a white ball is chosen out from pot $i\text{ and }{R}_i$ then be an occasion wherein a red ball is picked as from vase $i,i\in [1,2]$

Step 2: Probability (part a)

a) 

$P\left(W_2\right)=P\left(W_2\mid W_1\right)P\left(W_1\right)+P\left(W_2\mid R_1\right)P\left(R_1\right)=\frac{2}{3}\times \frac{2}{6}+\frac{1}{3}\times \frac{4}{6}=\frac{4}{9}$

  

Step 3: C hance (part b)

b) $P\left(W_1\mid W_2\right)=\frac{P\left(W_2\mid W_1\right)P\left(W_1\right)}{P\left(W_2\right)}=\frac{\frac{2}{3}\times \frac{2}{6}}{\frac{4}{9}}=\frac{1}{2}$