Question

Suppose there are three urns containing 2 white and 3 black balls: 3 white and 2 black balls, and 4 white and one black ball respectively. There is equal probability of each urn being chosen. One ball is drawn from an urn chosen at random. The probability that a white ball is drawn is $\frac{k}{15}$. Find the value of $k$.

Answer 1

Let $A_i(i=1,2,3)$ be the event that ith urn is chosen and $B$ the event that a white ball is drawn.
Since all the urns are equally likey to be selected, we have
$P\left(A_1\right)=P\left(A_2\right)=P\left(A_3\right)=\frac{1}{3}$
and $P\left(B/A_1\right),P\left(B/A_2\right)=\frac{3}{5},P\left(B/A_3\right)=\frac{4}{5}$
Hence $P\left(B\right)=P\left(A_1\right)P\left(B/A_2\right)P\left(B/A_2\right)P\left(A_3\right)P\left(B/A_3\right)$
$=\frac{1}{3}.\frac{2}{5}+\frac{1}{3}.\frac{3}{5}+\frac{1}{3}.\frac{4}{5}=\frac{3}{5}$