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 :
exactly one success

Answer

To solve the problem of finding the probability of getting exactly 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 1: Identify the total number of balls and the number of successes.
- There are 25 balls in total.
- The odd-numbered balls (which are considered successes) are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25.
- Therefore, the number of odd balls = 13.
- The number of even balls = 25 - 13 = 12.

### Step 2: Calculate the probability of success and failure.
- The probability of drawing an odd ball (success) = Number of odd balls / Total number of balls = 13 / 25.
- The probability of drawing an even ball (failure) = Number of even balls / Total number of balls = 12 / 25.

### Step 3: Determine the scenarios for exactly one success.
When drawing two balls, we can have exactly one success in two possible scenarios:
1. The first ball is odd (success) and the second ball is even (failure).
2. The first ball is even (failure) and the second ball is odd (success).

### Step 4: Calculate the probability for each scenario.
1. Probability of first ball being odd and second ball being even:
- P(Odd, Even) = (Probability of odd) * (Probability of even) = (13/25) * (12/25).

2. Probability of first ball being even and second ball being odd:
- P(Even, Odd) = (Probability of even) * (Probability of odd) = (12/25) * (13/25).

### Step 5: Combine the probabilities of both scenarios.
- Total probability of getting exactly one success:
\[ P(\text{Exactly one success}) = P(Odd, Even) + P(Even, Odd) \]
\[ = \left(\frac{13}{25} \cdot \frac{12}{25}\right) + \left(\frac{12}{25} \cdot \frac{13}{25}\right) \]
\[ = 2 \cdot \left(\frac{13}{25} \cdot \frac{12}{25}\right) \]
\[ = 2 \cdot \frac{156}{625} = \frac{312}{625}. \]

### Final Answer:
The probability of getting exactly one success when drawing two balls from the urn is \(\frac{312}{625}\).

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