A card is drawn from a box, which have cards marked with numbers 2 to 101, mixed thoroughly. One card is drawn from the box. What is the probability that the card taken out bears a number which is a perfect cube?

Options

A. `(1)/(20)`
B. `(7)/(100)`
C. `(9)/(100)`
D. `(3)/(100)`

Answer

To solve the problem of finding the probability that a card drawn from a box, which has cards marked with numbers from 2 to 101, bears a number that is a perfect cube, we can follow these steps:

### Step 1: Identify the Total Outcomes
The total outcomes are the total number of cards in the box. The cards are marked with numbers from 2 to 101.

- The range of numbers is from 2 to 101, inclusive.
- To find the total number of cards, we calculate:
\[ \text{Total numbers} = 101 - 2 + 1 = 100 \]

### Step 2: Identify the Favorable Outcomes
Next, we need to find the numbers between 2 and 101 that are perfect cubes. A perfect cube is a number that can be expressed as \( n^3 \), where \( n \) is an integer.

- The perfect cubes within the range can be calculated as follows:
- \( 1^3 = 1 \) (not included since it is less than 2)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( 5^3 = 125 \) (not included since it is greater than 101)

Thus, the perfect cubes between 2 and 101 are:
- 8
- 27
- 64

This gives us a total of 3 favorable outcomes.

### Step 3: Calculate the Probability
The probability \( P \) of drawing a card that bears a perfect cube is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]

Substituting the values we found:
\[ P = \frac{3}{100} \]

### Conclusion
The probability that the card drawn bears a number which is a perfect cube is:
\[ \frac{3}{100} \]