* 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,= 30, p= 0.04, x= 2
P(2) =
.1924
(Do not round until the final answer. Then round to four decimal places as needed.)

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

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

Where:
- \( n \) is the number of trials,
- \( x \) is the number of successes,
- \( p \) is the probability of success on each trial,
- \( \binom{n}{x} \) is the binomial coefficient, which is calculated as:

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

Now, plug in the given values:
- \( n = 30 \),
- \( p = 0.04 \),
- \( x = 2 \).

We first calculate the binomial coefficient:

\[
\binom{30}{2} = \frac{30!}{2!(30 - 2)!} = \frac{30 \times 29}{2 \times 1} = 435
\]

Now, compute the probability:

\[
P(2) = 435 \times (0.04)^2 \times (1 - 0.04)^{30 - 2}
\]
\[
P(2) = 435 \times 0.0016 \times (0.96)^{28}
\]
\[
P(2) = 435 \times 0.0016 \times 0.3725
\]
\[
P(2) \approx 435 \times 0.000596 = 0.25926
\]

Finally, rounding to four decimal places:

\[
P(2) \approx 0.2593
\]

Thus, the probability of 2 successes is approximately **0.2593**.

Would you like a breakdown of the steps or have any further questions?

Here are 5 questions to expand your understanding:
1. How does the binomial distribution change with different probabilities \( p \)?
2. What is the expected number of successes in this scenario?
3. How would the probability change if we wanted 3 successes instead of 2?
4. How do you calculate cumulative probabilities for \( P(x \leq 2) \)?
5. What happens to the binomial distribution as \( n \) increases while keeping \( p \) constant?

**Tip:** When working with binomial probabilities, it's helpful to familiarize yourself with using a scientific calculator or software for large factorials.

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 = 30, p = 0.04, x = 2. P(2) = ? (Do not round until the final answer. Then round to four decimal places as needed.)

Learn how to calculate the probability of 2 successes in a binomial probability experiment with n = 30 trials and a success probability of p = 0.04. Follow the step-by-step solution using the binomial formula, combinatorics, and detailed calculations. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation