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

Answer 1

Answer: (A)

Let $B_1$ be the event where $Box-I$ is selected. $\&B_2\to$ where box-II selected
$P\left(B_1\right)=P\left(B_2\right)=\frac{1}{2}$
Let $E$ be the event where selected card is non prime.
For $B_1:$ Prime numbers
$\{2,3,5,7,11,13,17,19,23,29\}$
For $B_2:$ Prime numbers :
$\{31,37,41,43,47\}$
$P(E)=P\left(B_1\right)\times P\left(\frac{E}{B_1}\right)+P\left(B_2\right)P\left(\frac{E}{B_2}\right)$
$=\frac{1}{2}\times \frac{20}{30}+\frac{1}{2}\times \frac{15}{20}$
Required probability
$P\left(\frac{B_1}{E}\right)=\frac{\frac{1}{2}\times \frac{20}{30}}{\frac{1}{2}\times \frac{20}{30}+\frac{1}{2}\times \frac{15}{20}}$
$=\frac{\frac{2}{3}}{\frac{2}{3}+\frac{3}{4}}=\frac{8}{17}$