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To solve this problem, we use the binomial probability formula, given by:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

where:
- \( n = 4 \): the number of trials,
- \( p = 0.25 \): the probability of success on each trial,
- \( k = 2 \): the number of successes we're interested in.

### Step-by-Step Solution:

1. **Calculate \(\binom{n}{k}\)**:
   \[
   \binom{n}{k} = \frac{n!}{k!(n - k)!} = \frac{4!}{2!(4 - 2)!} = \frac{4 \times 3}{2 \times 1} = 6
   \]

2. **Apply the binomial probability formula**:
   Substitute the values into the formula:
   \[
   P(X = 2) = 6 \times (0.25)^2 \times (1 - 0.25)^{4 - 2}
   \]

3. **Compute the probability**:
   - \( (0.25)^2 = 0.0625 \)
   - \( (1 - 0.25) = 0.75 \)
   - \( 0.75^{2} = 0.5625 \)

   So,
   \[
   P(X = 2) = 6 \times 0.0625 \times 0.5625 = 0.2109375
   \]

4. **Round the answer to three decimal places**:
   \[
   P(X = 2) \approx 0.211
   \]

### Answer:
The probability that the experiment results in exactly 2 successes is **0.211**.

Would you like more details or have any questions?

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### Related Questions:
1. How would the probability change if the number of trials \( n \) was increased to 6 with the same probability of success?
2. What is the probability of getting exactly 1 success in this scenario?
3. How does the probability change if the success probability \( p \) is increased to 0.5?
4. What is the probability of having 0 successes with \( n = 4 \) and \( p = 0.25 \)?
5. How can we calculate the cumulative probability of getting at most 2 successes?

### Tip:
When solving binomial problems, double-check the values of \( n \), \( k \), and \( p \) to ensure they’re substituted correctly, as small errors in setup can lead to incorrect answers.

A binomial experiment with probability of success p=0.25 and n=4 trials is conducted. What is the probability that the experiment results in exactly 2 successes?

Learn how to calculate the probability of getting exactly 2 successes in a binomial experiment with 4 trials and a 25% success rate. This problem covers the binomial probability formula and key combinatorial calculations. Suitable for high school students in Grades 10-12.

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