A number is taken from 1 to 100. The probability that it is a prime number :

Question

Accepted Answer

To determine the probability that a randomly chosen number from 1 to 100 is a prime number, we need to first identify how many prime numbers exist within this range.

1. **Identifying Prime Numbers from 1 to 100:**
   - Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
   - The prime numbers between 1 and 100 are: 
     2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
   - There are 25 prime numbers between 1 and 100.

2. **Calculating the Probability:**
   - The total number of possible outcomes (numbers from 1 to 100) is 100.
   - The number of favorable outcomes (prime numbers) is 25.
   - The probability $ P $ of selecting a prime number is given by the ratio of favorable outcomes to the total number of outcomes:
     $$
     P = \frac{\text{Number of prime numbers}}{\text{Total numbers}} = \frac{25}{100} = \frac{1}{4}
     $$

3. **Explanation of Why Option 3 is Correct:**
   - Option 3 states that the probability is $\frac{1}{4}$, which matches our calculated probability of $\frac{25}{100}$.
   - Therefore, Option 3 is the correct answer as it accurately represents the probability of selecting a prime number from 1 to 100.

4. **Explanation of Why Other Options are Incorrect:**
   - **Option 1: $\frac{1}{5}$**
     - This option suggests that there are 20 prime numbers (since $\frac{1}{5} \times 100 = 20$), which is incorrect because there are actually 25 prime numbers.
   - **Option 2: $\frac{3}{5}$**
     - This option implies there are 60 prime numbers (since $\frac{3}{5} \times 100 = 60$), which is far more than the actual count of 25.
   - **Option 4: $\frac{6}{25}$**
     - This option suggests there are 24 prime numbers (since $\frac{6}{25} \times 100 = 24$), which is close but still incorrect as the actual number is 25.

In conclusion, Option 3 is the correct choice because it accurately reflects the probability of selecting a prime number from 1 to 100 as $\frac{1}{4}$.