Question

A purse contains $2$ silver and $4$ copper coins. $A$ second purse contains $4$ silver and $3$ copper coins. If a coin is pulled at random from one of the two purses, what is the probability that it is a silver coin?

Answer 1

Consider the following events:

$E_1=$Selecting purse, $E_2=$Selecting second purse, $A=$Coin draw is silver coin.

We have, $P\left(E_1\right)=P\left(E_2\right)=\frac{1}{2},P\left(A/E_1\right)=\frac{2}{6},P\left(A/E_2\right)=\frac{4}{7}$

Required probability $=P\left(A\right)=P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)$.

$=\frac{1}{2}\times \frac{2}{6}+\frac{1}{2}\times \frac{4}{7}$

$=\frac{1}{6}+\frac{2}{7}$

$=\frac{7+12}{42}=\frac{19}{42}$