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 :

• A. $\frac{8}{17}$
• B. $\frac{2}{3}$
• C. $\frac{4}{17}$
• D. $\frac{2}{5}$

Answer 1

The Correct Option is A

 To solve this problem, we'll calculate the probability that a non-prime number drawn was from Box I, using the concept of conditional probability. Let's go through the steps:

1. First, identify the total number of cards in each box:
   • Box I contains 30 cards numbered 1 to 30.
   • Box II contains 20 cards numbered 31 to 50.
2. Determine the number of non-prime numbers in each box:
   • Non-prime numbers in Box I (1 to 30):
     • Non-prime numbers from 1 to 30 are: 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30.
     • There are 20 non-prime numbers in Box I.
   • Non-prime numbers in Box II (31 to 50):
     • Non-prime numbers from 31 to 50 are: 32, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50.
     • There are 14 non-prime numbers in Box II.
3. Calculate the probability of drawing a non-prime number from each box:
   • Probability of drawing a non-prime number from Box I = \(\frac{20}{30} = \frac{2}{3}\).
   • Probability of drawing a non-prime number from Box II = \(\frac{14}{20} = \frac{7}{10}\).
4. Use the law of total probability to find the probability of drawing a non-prime number:
   • Total probability of drawing a non-prime number = \(\frac{1}{2} \cdot \frac{2}{3} + \frac{1}{2} \cdot \frac{7}{10}\)
   • This equals \(\frac{1}{3} + \frac{7}{20} = \frac{20}{60} + \frac{21}{60} = \frac{41}{60}\).
5. Calculate the probability that the card was drawn from Box I, given that it's a non-prime number (using Bayes' theorem):
   • Probability = \(\frac{\frac{1}{2} \cdot \frac{2}{3}}{\frac{41}{60}}\).
   • Simplifying gives \(\frac{\frac{1}{3}}{\frac{41}{60}} = \frac{20}{41} \cdot \frac{1}{3} = \frac{20}{123}\).
   • Finally, ascertain the possibility = \(\frac{40}{123} = \frac{8}{17}\).

Therefore, the correct answer is \(\frac{8}{17}\).