An urn contains 25 balls numbered 1 to 25 .Suppose an odd number is considered a success Two balls are drawn from the urn with replacement.
Find the probability of getting at least one success.

Options

A. `(169)/(625)`
B. `(312)/(625)`
C. `(481)/(625)`
D. `(144)/(625)`

Answer

To find the probability of getting at least one success when drawing two balls from an urn containing 25 balls numbered from 1 to 25 (where an odd number is considered a success), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Balls**:
- The urn contains 25 balls numbered from 1 to 25.

2. **Determine the Number of Successes and Failures**:
- Odd numbers (successes): 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 → Total = 13 odd numbers.
- Even numbers (failures): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 → Total = 12 even numbers.

3. **Calculate the Probability of Success (p) and Failure (q)**:
- Probability of success (p) = Number of odd numbers / Total number of balls = 13/25.
- Probability of failure (q) = Number of even numbers / Total number of balls = 12/25.

4. **Find the Probability of Getting at Least One Success**:
- The probability of getting at least one success in two trials can be calculated using the complement rule:
\[ P(\text{at least one success}) = 1 - P(\text{no successes}) \]
- The probability of no successes (i.e., both draws are failures) when drawing two balls is:
\[ P(\text{no successes}) = q^2 = \left(\frac{12}{25}\right)^2 = \frac{144}{625} \]

5. **Calculate the Probability of At Least One Success**:
- Now, substituting into the complement formula:
\[ P(\text{at least one success}) = 1 - P(\text{no successes}) = 1 - \frac{144}{625} = \frac{625 - 144}{625} = \frac{481}{625} \]

### Final Answer:
The probability of getting at least one success when drawing two balls from the urn is \(\frac{481}{625}\).

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