Which of the following statements is CORRECT ?
Cards marked with the numbers 2 to 101 are put in a box and mixed thoroughly. One card is drawn from the box.
Statement-1 : Probability of odd prime numbers between 10 and 25 is `1/20`.
Statement-2 : Probability of perfect cube is `1/25`

Options

A. Only Statement-1
B. Only Statement-2
C. Both Statement-1 and Statement-2
D. Neither Statement-1 nor Statement-2

Answer

To determine which of the statements is correct, we will analyze both statements step by step.

### Step 1: Analyze Statement 1
**Statement-1:** Probability of odd prime numbers between 10 and 25 is \( \frac{1}{20} \).

1. **Identify the total number of cards:** The cards are numbered from 2 to 101. Therefore, the total number of cards is:
\[ 101 - 2 + 1 = 100 \]

2. **Identify odd prime numbers between 10 and 25:** The odd prime numbers in this range are:
- 11
- 13
- 17
- 19
- 23

Thus, the odd prime numbers between 10 and 25 are: 11, 13, 17, 19, and 23. This gives us a total of 5 odd prime numbers.

3. **Calculate the probability:** The probability of drawing an odd prime number is given by the formula:
\[ P(\text{odd prime}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{100} = \frac{1}{20} \]

### Conclusion for Statement 1:
Statement 1 is correct as the probability of odd prime numbers between 10 and 25 is indeed \( \frac{1}{20} \).

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### Step 2: Analyze Statement 2
**Statement-2:** Probability of perfect cubes is \( \frac{1}{25} \).

1. **Identify perfect cubes between 2 and 101:** The perfect cubes are:
- \( 1^3 = 1 \) (not in the range)
- \( 2^3 = 8 \) (not in the range)
- \( 3^3 = 27 \) (in the range)
- \( 4^3 = 64 \) (in the range)
- \( 5^3 = 125 \) (not in the range)

The perfect cubes between 2 and 101 are: 27 and 64. Thus, there are 2 perfect cubes.

2. **Calculate the probability:** The probability of drawing a perfect cube is given by:
\[ P(\text{perfect cube}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{100} = \frac{1}{50} \]

### Conclusion for Statement 2:
Statement 2 is incorrect as the probability of perfect cubes is \( \frac{1}{50} \), not \( \frac{1}{25} \).

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### Final Conclusion:
Since Statement 1 is correct and Statement 2 is incorrect, the correct answer is that **only Statement 1 is true**.

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