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 $\mathrm{\Pi }$?

Short 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-by-step solution

 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 $i,i\in [1,2]$

Probability (part a)

a)

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

Chance (part b)

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