Mathematics - Probability Question with Solution | TestHub

Let $E_1,E_2$ and $A$ be the events defined as follows:
${\mathrm{E}}_1=$ red ball is transferred from bag $\mathrm{P}$ to bag $\mathrm{Q}$
${\mathrm{E}}_2=$ blue ball is transferred from bag $\mathrm{P}$ to bag $\mathrm{Q}$
$\mathrm{A}=$ the ball drawn from bag $\mathrm{Q}$ is blue
As the bag $\mathrm{P}$ contains 6 red and 4 blue balls,
$P\left(E_1\right)=\frac{6}{10}=\frac{3}{5}\text{ and }P\left(E_2\right)=\frac{4}{10}=\frac{2}{5}$
Note that $E_1$ and $E_2$ are mutually exclusive and exhaustive events.
When $E_1$ has occurred i.e., a red ball has already been transferred from bag $P$ to $Q$, then bag $Q$ will contain 6 red and 6 blue balls, So, $P\left(A\mid E_1\right)=$$\frac{6}{12}=\frac{1}{2}$
When $E_2$ has occurred i.e., a blue ball has already been transferred from bag $P$ to $Q$, then bag $Q$ will contain 5 red and 7 blue balls, So, $P\left(A\mid E_2\right)=$$\frac{7}{12}$
By using law of total probability, we get
$P(A)=P\left(E_1\right)P\left(A\mid E_1\right)+P\left(E_2\right)P\left(A\mid E_2\right)$
$=\frac{3}{5}\times \frac{1}{2}+\frac{2}{5}\times \frac{7}{12}=\frac{8}{15}$