Box-1 contains 30 cards marked from 1 to 30 and Box-2 contains 20 cards amked from 31 to 50. A box is selected and a card is drawn. If the number on the card is non-prime then what is the probability that it came from Box 1.

Options

A. 4/17
B. 6/17
C. 7/17
D. 8/17

Answer

To solve the problem, we will use Bayes' Theorem to find the probability that a non-prime card drawn came from Box 1.

### Step-by-Step Solution:

1. **Identify the Total Cards in Each Box:**
- Box 1 contains cards numbered from 1 to 30 (30 cards).
- Box 2 contains cards numbered from 31 to 50 (20 cards).

2. **Determine the Total Number of Non-Prime Cards in Each Box:**
- **Box 1 (1 to 30):**
- Prime numbers in Box 1: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (Total: 10 prime numbers).
- Non-prime numbers = Total cards - Prime numbers = 30 - 10 = 20 non-prime numbers.

- **Box 2 (31 to 50):**
- Prime numbers in Box 2: 31, 37, 41, 43, 47 (Total: 5 prime numbers).
- Non-prime numbers = Total cards - Prime numbers = 20 - 5 = 15 non-prime numbers.

3. **Calculate the Total Probability of Drawing a Non-Prime Card:**
- Probability of choosing Box 1 = P(Box 1) = 1/2.
- Probability of drawing a non-prime card from Box 1 = P(Non-prime | Box 1) = 20/30 = 2/3.
- Probability of choosing Box 2 = P(Box 2) = 1/2.
- Probability of drawing a non-prime card from Box 2 = P(Non-prime | Box 2) = 15/20 = 3/4.

4. **Calculate the Total Probability of Drawing a Non-Prime Card:**
- Total probability of drawing a non-prime card, P(Non-prime) = P(Box 1) * P(Non-prime | Box 1) + P(Box 2) * P(Non-prime | Box 2)
- P(Non-prime) = (1/2 * 20/30) + (1/2 * 15/20)
- P(Non-prime) = (1/2 * 2/3) + (1/2 * 3/4)
- P(Non-prime) = (1/3) + (3/8)
- To add these fractions, find a common denominator (24):
- (1/3) = 8/24
- (3/8) = 9/24
- P(Non-prime) = 8/24 + 9/24 = 17/24.

5. **Use Bayes' Theorem to Find the Desired Probability:**
- We want to find P(Box 1 | Non-prime) = (P(Non-prime | Box 1) * P(Box 1)) / P(Non-prime).
- P(Box 1 | Non-prime) = (2/3 * 1/2) / (17/24).
- P(Box 1 | Non-prime) = (1/3) / (17/24).
- P(Box 1 | Non-prime) = (1/3) * (24/17) = 8/17.

### Final Answer:
The probability that the non-prime card came from Box 1 is **8/17**.