Box I contains 30 cards numbered

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 :

1. (1) $\frac{2}{3}$

2. (2) $\frac{8}{17}$

3. (3) $\frac{4}{17}$

4. (4) $\frac{2}{5}$

Correct Option: , 2

Solution:

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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