Natural numbers 4 to 15 are written on different slips (one number on one slip), kept in a box and mixed well. Renu pick up one slip from the box without looking into it. What is the probability that this slip bears a prime number?

Options

A. `(1)/(3)`
B. `(5)/(12)`
C. `(2)/(5)`
D. `(3)/(11)`

Answer

To solve the problem of finding the probability that Renu picks a slip with a prime number from the numbers 4 to 15, we can follow these steps:

### Step 1: Identify the total numbers
The natural numbers from 4 to 15 are:
\[ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 \]

### Step 2: Count the total numbers
To find the total count of numbers from 4 to 15:
- The range is from 4 to 15, which includes both endpoints.
- Total numbers = \( 15 - 4 + 1 = 12 \)

### Step 3: Identify the prime numbers within the range
Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The prime numbers between 4 and 15 are:
- 5 (divisors: 1, 5)
- 7 (divisors: 1, 7)
- 11 (divisors: 1, 11)
- 13 (divisors: 1, 13)

### Step 4: Count the prime numbers
From our list, the prime numbers are:
\[ 5, 7, 11, 13 \]
- Total prime numbers = 4

### Step 5: Calculate the probability
The probability \( P \) of picking a prime number is given by the formula:
\[ P(\text{prime}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
Substituting the values we found:
\[ P(\text{prime}) = \frac{4}{12} \]

### Step 6: Simplify the fraction
To simplify \( \frac{4}{12} \):
\[ \frac{4}{12} = \frac{1}{3} \]

### Conclusion
The probability that Renu picks a slip with a prime number is:
\[ \frac{1}{3} \]