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 :
Solution
Let B<sub>1</sub> be the event where Box-I is selected.
<br><br>And B<sub>2</sub> be the event where Box-II is selected.
<br><br>P(B<sub>1</sub>) = P(B<sub>2</sub>) = ${1 \over 2}$
<br><br>Let E be the event where selected card is non prime.
<br><br>For B<sub>1</sub> : Prime numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
<br><br>For B<sub>2</sub> : Prime numbers: {31, 37, 41, 43, 47}
<br><br>P(E) = P(B<sub>1</sub>) $\times$ $P\left( {{E \over {{B_1}}}} \right)$ + P(B<sub>2</sub>) $\times$ $P\left( {{E \over {{B_2}}}} \right)$
<br><br>= ${1 \over 2} \times {{20} \over {30}}$ + ${1 \over 2} \times {{15} \over {20}}$
<br><br>Required probability :
<br><br>$P\left( {{{{B_1}} \over E}} \right)$ = $${{P\left( {{B_2}} \right).P\left( {{E \over {{B_1}}}} \right)} \over {P\left( E \right)}}$$
<br><br>= $${{{1 \over 2} \times {{20} \over {30}}} \over {{1 \over 2} \times {{20} \over {30}} + {1 \over 2}{{15} \over {20}}}}$$
<br><br>= ${{{2 \over 3}} \over {{2 \over 3} + {3 \over 4}}}$
<br><br>= ${8 \over {17}}$
About this question
Subject: Mathematics · Chapter: Probability · Topic: Classical and Axiomatic Probability
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