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

A silver coin can be drawn in two mutually exclusive ways:
(I) Selecting purse I and then drawing a silver coin from it
(II) Selecting purse II and then drawing a silver coin from it
Let E₁, E₂ and A be the events as defined below:
E₁ = Selecting purse I
E₂ = Selecting purse II
A = Drawing a silver coin
It is given that one of the purses is selected randomly

$$\therefore P\left(E_1\right)=\frac{1}{2}$$

$$P\left(E_2\right)=\frac{1}{2}$$

$$\text{ Now },$$

$$P\left(A/E_1\right)=\frac{2}{6}=\frac{1}{3}$$

$$P\left(A/E_2\right)=\frac{4}{7}$$

$$\text{ Using the law of total probability, we get}$$

$$\text{ Required probability }=P\left(A\right)=P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)$$

$$=\frac{1}{2}\times \frac{1}{3}+\frac{1}{2}\times \frac{4}{7}$$

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

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