Cards bearing numbers 12, 13, 14, ... , 70, 71 (one number on one card) are kept in a box and then a card is drawn from the box without looking into it. What is the probability that the card drawn has the number which is not a prime ?

Question

Accepted Answer

To determine the probability that the card drawn has a number which is not a prime, we need to follow these steps:

1. **Identify the Range of Numbers:**
   The cards have numbers ranging from 12 to 71. This gives us a total of:
   $$
   71 - 12 + 1 = 60 \text{ numbers}
   $$

2. **Identify Prime Numbers in the Range:**
   A prime number is a number greater than 1 that has no divisors other than 1 and itself. We need to find all the prime numbers between 12 and 71.

The prime numbers in this range are: 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71. This gives us a total of 15 prime numbers.

3. **Calculate Non-Prime Numbers:**
   The non-prime numbers are simply the total numbers minus the prime numbers:
   $$
   60 - 15 = 45 \text{ non-prime numbers}
   $$

4. **Calculate the Probability:**
   The probability that a card drawn has a number which is not a prime is the ratio of non-prime numbers to the total numbers:
   $$
   \frac{45}{60} = \frac{3}{4}
   $$

Thus, the probability that the card drawn has a number which is not a prime is $\frac{3}{4}$, which corresponds to Option 4.

**Explanation of Incorrect Options:**

- **Option 1: $\frac{45}{59}$**
  - This option is incorrect because it suggests a total of 59 numbers, which is not the case. The total number of cards is 60, not 59.

- **Option 2: $\frac{1}{4}$**
  - This option is incorrect because it implies that only a quarter of the numbers are non-prime, which is not true. In fact, three-quarters of the numbers are non-prime.

- **Option 3: $\frac{15}{59}$**
  - This option is incorrect because it suggests a probability based on 59 numbers, and it also incorrectly represents the count of non-prime numbers. The correct count of non-prime numbers is 45, not 15.

Therefore, the correct answer is Option 4: $\frac{3}{4}$.