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This problem is asking for the probability of getting exactly 3 successes in a binomial experiment where \( p = 0.25 \) (the probability of success) and \( n = 5 \) (the number of trials). To solve this, we can use the **binomial probability formula**:

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

Where:
- \( n = 5 \) (the number of trials),
- \( k = 3 \) (the number of successes),
- \( p = 0.25 \) (the probability of success),
- \( 1 - p = 0.75 \) (the probability of failure).

### Step 1: Calculate the binomial coefficient \( \binom{n}{k} \)
The binomial coefficient \( \binom{n}{k} \) is calculated as:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

For \( n = 5 \) and \( k = 3 \):

\[
\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10
\]

### Step 2: Plug values into the binomial probability formula
Now, substitute the values into the formula:

\[
P(X = 3) = 10 \times (0.25)^3 \times (0.75)^2
\]

### Step 3: Simplify the expression
\[
(0.25)^3 = 0.015625, \quad (0.75)^2 = 0.5625
\]
\[
P(X = 3) = 10 \times 0.015625 \times 0.5625 = 10 \times 0.0087890625 = 0.087890625
\]

### Step 4: Round the result
Rounding to three decimal places:

\[
P(X = 3) \approx 0.088
\]

So, the probability of getting exactly 3 successes is **0.088**.

---

Would you like more details or have any questions about the process? Here are some related questions you might explore:

1. How do you calculate the probability of fewer than 3 successes?
2. How does changing \( p \) (probability of success) affect the outcome in a binomial distribution?
3. What is the difference between a binomial distribution and a normal distribution?
4. How do you calculate the expected number of successes in a binomial experiment?
5. Can the binomial probability formula be used for very large \( n \) values?

**Tip:** Remember to double-check the rounding in binomial probability problems, as this often leads to small errors in the final answer!

A binomial experiment with probability of success p = 0.25 and n = 5 trials is conducted. What is the probability that the experiment results in exactly 3 successes? Do not round your intermediate computations, and round your answer to three decimal places.

This problem demonstrates how to calculate the probability of getting exactly 3 successes in a binomial experiment with p = 0.25 and n = 5. Using the binomial probability formula and key probability concepts, it offers a detailed step-by-step explanation.

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