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, we will follow these steps:

### Step 1: Determine the total number of balls and the number of odd-numbered balls.
The urn contains a total of 25 balls, numbered from 1 to 25. Among these, the odd-numbered balls are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, and 25.

**Total number of odd balls = 13.**

### Step 2: Calculate the probability of drawing an odd-numbered ball on the first draw.
The probability of drawing an odd-numbered ball on the first draw is given by the ratio of the number of odd balls to the total number of balls:

\[ P(\text{first odd}) = \frac{\text{Number of odd balls}}{\text{Total number of balls}} = \frac{13}{25} \]

### Step 3: Calculate the probability of drawing an odd-numbered ball on the second draw.
Since the balls are drawn with replacement, the total number of balls remains the same for the second draw. Therefore, the probability of drawing an odd-numbered ball on the second draw is also:

\[ P(\text{second odd}) = \frac{13}{25} \]

### Step 4: Calculate the combined probability of both events.
Since the draws are independent (because of replacement), the combined probability of both draws resulting in odd-numbered balls is the product of the individual probabilities:

\[ P(\text{both odd}) = P(\text{first odd}) \times P(\text{second odd}) = \frac{13}{25} \times \frac{13}{25} = \left(\frac{13}{25}\right)^2 \]

### Step 5: Final calculation.
Now we can compute the final probability:

\[ P(\text{both odd}) = \frac{169}{625} \]

Thus, the probability that both the numbers on the balls drawn are odd is:

\[ \frac{169}{625} \]

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