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

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

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

\[\text{ Now } , \]

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

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

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

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

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

\[ = \frac{1}{6} + \frac{2}{7}\]

\[ = \frac{7 + 12}{42} = \frac{19}{42}\]