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 :
Correct Option: , 2
Let $B_{1}$ and $B_{2}$ be the boxes and $N$ be the number of
non-prime number.
and $P$ (non-prime number)
$=P\left(B_{1}\right) \times P\left(\frac{N}{B_{1}}\right)+P\left(B_{2}\right) \times P\left(\frac{N}{B_{2}}\right)$
$=\frac{1}{2} \times \frac{20}{30}+\frac{1}{2} \times \frac{15}{20}$
So,
$P\left(\frac{B_{1}}{N}\right)=\frac{P\left(B_{1}\right) \times P\left(\frac{N}{B_{1}}\right)}{P\left(B_{1}\right) \times P\left(\frac{N}{B_{1}}\right)+P\left(B_{2}\right) \times P\left(\frac{N}{B_{2}}\right)}$
$=\frac{\frac{1}{2} \times \frac{20}{30}}{\frac{1}{2} \times \frac{20}{30}+\frac{1}{2} \times \frac{15}{20}}=\frac{\frac{1}{3}}{\frac{1}{3}+\frac{15}{40}}=\frac{8}{17}$
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