Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50.

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 :

Accepted Answer

Let B1 be the event where Box-I is selected.

And B2 be the event where Box-II is selected.

P(B1) = P(B2) = $${1 \over 2}$$

Let E be the event where selected card is non prime.

For B1 : Prime numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

For B2 : Prime numbers: {31, 37, 41, 43, 47}

P(E) = P(B1) $$ \times $$ $$P\left( {{E \over {{B_1}}}} \right)$$ +  P(B2) $$ \times $$ $$P\left( {{E \over {{B_2}}}} \right)$$

= $${1 \over 2} \times {{20} \over {30}}$$ + $${1 \over 2} \times {{15} \over {20}}$$

Required probability :

$$P\left( {{{{B_1}} \over E}} \right)$$ = $${{P\left( {{B_2}} \right).P\left( {{E \over {{B_1}}}} \right)} \over {P\left( E \right)}}$$

= $${{{1 \over 2} \times {{20} \over {30}}} \over {{1 \over 2} \times {{20} \over {30}} + {1 \over 2}{{15} \over {20}}}}$$

= $${{{2 \over 3}} \over {{2 \over 3} + {3 \over 4}}}$$

= $${8 \over {17}}$$

Solutions · 1