A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.
nequals50​, pequals0.04​, xequals2

In a binomial probability experiment, the probability of obtaining exactly \( x \) successes in \( n \) independent trials, where the probability of success on each trial is \( p \), is given by the binomial probability formula:

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

Where:
- \( n \) is the number of trials,
- \( x \) is the number of successes,
- \( p \) is the probability of success on a single trial,
- \( \binom{n}{x} \) is the binomial coefficient, which represents the number of ways to choose \( x \) successes from \( n \) trials, and is calculated as:

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

### Given parameters:
- \( n = 50 \)
- \( p = 0.04 \)
- \( x = 2 \)

### Step 1: Calculate the binomial coefficient \( \binom{50}{2} \)

\[
\binom{50}{2} = \frac{50!}{2!(50 - 2)!} = \frac{50 \times 49}{2 \times 1} = 1225
\]

### Step 2: Compute \( p^x \) and \( (1 - p)^{n - x} \)

- \( p^2 = (0.04)^2 = 0.0016 \)
- \( (1 - p)^{50 - 2} = (0.96)^{48} \approx 0.1170 \)

### Step 3: Calculate the probability

\[
P(X = 2) = 1225 \times 0.0016 \times 0.1170 \approx 0.2295
\]

Thus, the probability of exactly 2 successes in 50 trials, with a success probability of 0.04 on each trial, is approximately **0.2295**.

Would you like more details on any of the steps, or do you have further questions?

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Here are 5 related questions you can explore:
1. How does changing \( p \) (probability of success) affect the binomial probability?
2. What is the probability of getting exactly 1 success with the same parameters?
3. How can you use the cumulative binomial probability to find the probability of getting at most 2 successes?
4. What is the expected number of successes in this experiment?
5. How would the probability change if \( n \) were 100 instead of 50?

**Tip:** For large \( n \) and small \( p \), the binomial distribution can be approximated by a Poisson distribution.

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 50, p = 0.04, x = 2

Discover how to compute the probability of exactly 2 successes in 50 independent trials with a success probability of 0.04. This problem involves key probability concepts such as the binomial distribution, binomial coefficient, and the binomial probability formula. Suitable for Grades 10-12 students.

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