An urn contains 25 balls numbered from 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

Answer

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

### Step-by-Step Solution:

1. **Identify the Total Number of Balls and Success Criteria**:
- The urn contains 25 balls numbered from 1 to 25.
- An odd number is considered a success. The odd numbers between 1 and 25 are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25.
- There are 13 odd numbers.

**Hint**: Count the odd numbers from 1 to 25 to determine the number of successes.

2. **Calculate the Probability of Success (p)**:
- The probability of drawing an odd number (success) is given by:
\[ p = \frac{\text{Number of odd balls}}{\text{Total number of balls}} = \frac{13}{25} \]

**Hint**: Use the formula for probability, which is the number of favorable outcomes divided by the total outcomes.

3. **Calculate the Probability of Failure (q)**:
- The probability of drawing an even number (failure) is:
\[ q = 1 - p = 1 - \frac{13}{25} = \frac{12}{25} \]

**Hint**: Remember that the sum of probabilities of success and failure must equal 1.

4. **Determine the Probability of No Success**:
- We want to find the probability of getting at least one success when drawing 2 balls. It is easier to first calculate the probability of getting no successes (i.e., both balls drawn are even).
- The probability of getting no successes (0 successes) in 2 trials is given by:
\[ P(\text{no success}) = q^n = \left(\frac{12}{25}\right)^2 \]

**Hint**: Use the formula for the probability of independent events for multiple trials.

5. **Calculate the Probability of At Least One Success**:
- The probability of getting at least one success is given by:
\[ P(\text{at least one success}) = 1 - P(\text{no success}) = 1 - \left(\frac{12}{25}\right)^2 \]
- Now calculate:
\[ P(\text{at least one success}) = 1 - \frac{144}{625} = \frac{625 - 144}{625} = \frac{481}{625} \]

**Hint**: Subtract the probability of no successes from 1 to find the probability of at least one success.

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