Box-l 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

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$2 / 3$

$8 / 17$

$2 / 5$

$4 / 17$

answer is B.

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Detailed Solution

Let A₁, A₂ be the events of selecting a Box from Box-I and Box-II respectively.

Then $P (A_{1}) = \frac{1}{2}, P (A_{2}) = \frac{1}{2}$

Let E be the event that the number on the selected card is non prime.

Prime numbers from 1 to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Number of numbers between 1 to 30 which are not prime= 20

Number of numbers between 31 to 50 which are not prime= 15

then $P (E ∣ A_{1}) = \frac{2}{3}$ and $P (E ∣ A_{2}) = \frac{3}{4}$

Selecting non-prime numbered card from box-1 is $P (A_{1} ∣ E) = \frac{P (A_{1}) P (E ∣ A_{1})}{P (A_{1}) P (E ∣ A_{1}) + P (A_{2}) P (E ∣ A_{2})} = \frac{(\frac{1}{2}) (\frac{2}{3})}{(\frac{1}{2}) (\frac{2}{3}) + (\frac{1}{2}) (\frac{3}{4})} = \frac{2 / 3}{2 / 3 + 3 / 4} = \frac{8}{17}$