An urn contains 25 balls numbered 1 to 25. Two balls drawn one at a time with replacement. The probability that both the numbers on the balls are odd is

Answer

To solve the problem of finding the probability that both balls drawn from an urn containing 25 balls numbered 1 to 25 are odd, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Balls**:
The urn contains balls numbered from 1 to 25. Therefore, the total number of balls is:
\[ \text{Total Balls} = 25 \]

2. **Identify the Odd Numbers**:
The odd numbers between 1 and 25 are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, and 25. Counting these, we find:
\[ \text{Odd Numbers} = 13 \]

3. **Calculate the Probability of Drawing an Odd Number**:
The probability of drawing one odd number from the urn is given by the ratio of the number of odd balls to the total number of balls:
\[ P(\text{Odd}) = \frac{\text{Number of Odd Balls}}{\text{Total Balls}} = \frac{13}{25} \]

4. **Drawing with Replacement**:
Since the balls are drawn with replacement, the total number of balls remains the same for each draw. Therefore, the probability of drawing an odd number remains:
\[ P(\text{Odd on 2nd Draw}) = \frac{13}{25} \]

5. **Calculate the Probability of Both Draws Being Odd**:
Since the draws are independent (due to replacement), the probability of both draws resulting in odd numbers is the product of the probabilities of each draw:
\[ P(\text{Both Odd}) = P(\text{Odd on 1st Draw}) \times P(\text{Odd on 2nd Draw}) = \frac{13}{25} \times \frac{13}{25} = \frac{169}{625} \]

### Final Answer:
Thus, the probability that both numbers on the balls drawn are odd is:
\[ \frac{169}{625} \]