
Question content area top

Part 1

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 equals 9n=9​, 

p equals 0.8p=0.8​, 

x less than or equals 3x≤3

Question content area bottom

Part 1

The probability of 

x less than or equals 3x≤3 

successes is 

enter your response here. 

​(Round to four decimal places as​ needed.)

In this problem, we are dealing with a **binomial probability experiment**, where:

- \( n = 9 \) (the number of trials),
- \( p = 0.8 \) (the probability of success in each trial),
- and we are asked to find the probability of having \( x \leq 3 \) (3 or fewer successes).

The probability mass function for a binomial distribution is:

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

where:

- \( \binom{n}{x} \) is the binomial coefficient (the number of ways to choose \( x \) successes from \( n \) trials),
- \( p^x \) is the probability of \( x \) successes,
- \( (1 - p)^{n - x} \) is the probability of \( n - x \) failures.

### To solve for \( P(X \leq 3) \), we need to calculate:

\[
P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]

Let's compute these probabilities step by step.

I'll calculate that now.The probability of having \( x \leq 3 \) successes in 9 independent trials, with a success probability of 0.8, is approximately **0.0031** (rounded to four decimal places).

Would you like further details on this, or do you have any other questions?

Here are five related questions for deeper understanding:
1. How would the probability change if \( p = 0.6 \) instead of 0.8?
2. What is the probability of exactly 4 successes in the same experiment?
3. How do you calculate the expected number of successes in a binomial experiment?
4. How would you compute the cumulative probability for \( x \leq 5 \)?
5. What is the standard deviation of this binomial distribution?

**Tip:** For binomial distributions, if the number of trials \( n \) increases, the distribution tends to look more like a normal distribution (Central Limit Theorem).

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 = 9, p = 0.8, x ≤ 3. What is the probability of x ≤ 3 successes? (Round to four decimal places as needed.)

Learn how to compute the binomial probability for n=9 independent trials, where p=0.8 and the number of successes is x ≤ 3. This tutorial includes key formulas and step-by-step calculations. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation