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 no success .

Options

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

Answer

To solve the problem step by step, we will calculate the probability of getting no success when drawing two balls from an urn containing 25 balls numbered 1 to 25, where odd numbers are considered a success.

### Step 1: Identify the total number of balls and classify them
- The urn contains 25 balls numbered from 1 to 25.
- Odd numbers (success) are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 (total: 13 odd numbers).
- Even numbers (failure) are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 (total: 12 even numbers).

### Step 2: Calculate the probabilities of success and failure
- Probability of success (drawing an odd number) \( P(success) = \frac{13}{25} \)
- Probability of failure (drawing an even number) \( P(failure) = \frac{12}{25} \)

### Step 3: Determine the probability of no success in two draws
Since we are drawing with replacement, the draws are independent. We want the probability of getting no successes (i.e., both draws yield even numbers).

Using the formula for the probability of no successes in n trials:
\[ P(no\ success) = P(failure)^n \]
where \( n = 2 \) (the number of draws).

### Step 4: Calculate the probability of no success
\[ P(no\ success) = P(failure)^2 = \left(\frac{12}{25}\right)^2 \]
Calculating this gives:
\[ P(no\ success) = \frac{12}{25} \times \frac{12}{25} = \frac{144}{625} \]

### Conclusion
The probability of getting no success (both draws resulting in even numbers) is:
\[ \frac{144}{625} \]