Question

Box I contains $30$ cards numbered $1$ to $30$ and Box II contains $20$ cards numbered $31$ to $50$. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is:

• A. $\frac{4}{17}$
• B. $\frac{8}{17}$
• C. $\frac{2}{5}$
• D. $\frac{2}{3}$

Answer 1

The correct option is B $\frac{8}{17}$

Box I $:1$ to $30$ numbered cards
Prime on box I

$\{2,3,5,7,11,13,17,19,23,29\}$

Box II $:31$ to $50$ numbered cards
Prime on box II

$\{31,37,41,43,47\}$

A : selected number on card is non - prime

$P\left(A\right)=P\left(I\right).P\left(\frac{A}{I}\right)+P\left(II\right).P\left(\frac{A}{11}\right)$

=$\frac{1}{2}\times \frac{20}{30}+\frac{1}{2}.\frac{15}{20}$
Now, $P\left(\frac{I}{A}\right)=\frac{P\left(II\right).P\left(\frac{A}{I}\right)}{P\left(A\right)}$

= $\frac{\frac{1}{2}.\frac{20}{30}}{\frac{1}{2}.\frac{20}{30}+\frac{1}{2}.\frac{15}{20}}=\frac{\frac{2}{3}}{\frac{2}{3}+\frac{3}{4}}=\frac{8}{17}$