Question

There are three boxes each containing 3 Pink and 5 Yellow balls and also there are 2 boxes each containing 4 Pink and 2 Yellow balls. A Yellow ball is selected at random Find the probability that Yellow ball is from a box of the first group?

Answer 1

$\Rightarrow$ Here, we can see there are total $5$ boxes.

$\Rightarrow$ $P\left(E_1\right)$ be the probability selecting three boxes.

$\therefore$ $P\left(E_1\right)=\frac{3}{5}$

$\Rightarrow$ $P\left(E_2\right)$ be the probability selecting other two boxes.

$\therefore$ $P\left(E_2\right)=\frac{2}{5}$

$\Rightarrow$ Probability of selecting yellow ball from three boxes $P\left(\frac{A}{E_1}\right)=\frac{5}{8}$

$\Rightarrow$ Probability of selecting yellow ball from two boxes $P\left(\frac{A}{E_2}\right)=\frac{2}{6}$

Now, the probability that yellow ball is from a box of first group $=\frac{P\left(E_1\right)\times P\left(\frac{A}{E_1}\right)}{P\left(E_1\right)\times P\left(\frac{A}{E_1}\right)+P\left(E_2\right)\times P\left(\frac{A}{E_2}\right)}$

$=\frac{\frac{3}{5}\times \frac{5}{8}}{\frac{3}{5}\times \frac{5}{8}+\frac{2}{5}\times \frac{2}{6}}$

$=\frac{45}{61}$