Write the probability that a number selected at random from the set of first 100 natural numbers is a cube.

Options

A. `(1)/(25)`
B. `(2)/(25)`
C. `(3)/(25)`
D. `(4)/(25)`

Answer

To find the probability that a number selected at random from the set of the first 100 natural numbers is a cube, we can follow these steps:

### Step 1: Identify the set of first 100 natural numbers
The first 100 natural numbers are the integers from 1 to 100.

### Step 2: Determine the cubes within this range
We need to find all the perfect cubes that are less than or equal to 100. A perfect cube is a number that can be expressed as \( n^3 \) where \( n \) is a natural number.

- The cube of 1 is \( 1^3 = 1 \)
- The cube of 2 is \( 2^3 = 8 \)
- The cube of 3 is \( 3^3 = 27 \)
- The cube of 4 is \( 4^3 = 64 \)
- The cube of 5 is \( 5^3 = 125 \) (which is greater than 100)

Thus, the perfect cubes in the first 100 natural numbers are: 1, 8, 27, and 64.

### Step 3: Count the number of cubes
From the previous step, we found that there are 4 perfect cubes (1, 8, 27, and 64) in the first 100 natural numbers.

### Step 4: Calculate the total number of outcomes
The total number of outcomes is the total number of natural numbers we are considering, which is 100.

### Step 5: Calculate the probability
The probability \( P \) of selecting a cube from the first 100 natural numbers is given by the formula:

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Substituting the values we found:

\[ P = \frac{4}{100} = \frac{1}{25} \]

### Conclusion
The probability that a number selected at random from the set of the first 100 natural numbers is a cube is \( \frac{1}{25} \).

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