JEE MAIN - Mathematics (2020 - 2nd September Morning Slot - No. 3)
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 :
$${8 \over {17}}$$
$${2 \over 3}$$
$${2 \over 5}$$
$${4 \over {17}}$$
Explanation
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 :
$$P\left( {{{{B_1}} \over E}} \right)$$ = $${{P\left( {{B_2}} \right).P\left( {{E \over {{B_1}}}} \right)} \over {P\left( E \right)}}$$
= $${{{1 \over 2} \times {{20} \over {30}}} \over {{1 \over 2} \times {{20} \over {30}} + {1 \over 2}{{15} \over {20}}}}$$
= $${{{2 \over 3}} \over {{2 \over 3} + {3 \over 4}}}$$
= $${8 \over {17}}$$
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 :
$$P\left( {{{{B_1}} \over E}} \right)$$ = $${{P\left( {{B_2}} \right).P\left( {{E \over {{B_1}}}} \right)} \over {P\left( E \right)}}$$
= $${{{1 \over 2} \times {{20} \over {30}}} \over {{1 \over 2} \times {{20} \over {30}} + {1 \over 2}{{15} \over {20}}}}$$
= $${{{2 \over 3}} \over {{2 \over 3} + {3 \over 4}}}$$
= $${8 \over {17}}$$
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