To solve the problem of finding the probability of getting exactly one success when drawing two balls from an urn containing 25 balls (numbered 1 to 25), where an odd number is considered a "success", we can follow these steps:
### Step 1: Determine the Total Number of Balls and Classify Them
The urn contains 25 balls numbered from 1 to 25. Among these:
- Odd numbers (successes): 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 (total of 13 odd numbers)
- Even numbers (failures): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 (total of 12 even numbers)
### Step 2: Calculate the Probability of Success and Failure
- 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
### Step 3: Use the Binomial Probability Formula
We are drawing 2 balls with replacement, and we want to find the probability of getting exactly one success (odd number). We can use the binomial probability formula:
\[
P(X = m) = \binom{n}{m} p^m q^{n-m}
\]
Where:
- \( n \) = number of trials (2 in this case)
- \( m \) = number of successes (1 in this case)
- \( p \) = probability of success (13/25)
- \( q \) = probability of failure (12/25)
### Step 4: Plug in the Values
Now we substitute the values into the formula:
\[
P(X = 1) = \binom{2}{1} \left(\frac{13}{25}\right)^1 \left(\frac{12}{25}\right)^{2-1}
\]
Calculating the binomial coefficient:
\[
\binom{2}{1} = 2
\]
Now substituting this into the equation:
\[
P(X = 1) = 2 \cdot \left(\frac{13}{25}\right) \cdot \left(\frac{12}{25}\right)
\]
### Step 5: Calculate the Final Probability
Now we compute the final probability:
\[
P(X = 1) = 2 \cdot \frac{13 \cdot 12}{25 \cdot 25} = \frac{2 \cdot 156}{625} = \frac{312}{625}
\]
### Final Answer
The probability of getting exactly one success when drawing two balls from the urn is:
\[
\frac{312}{625}
\]
