Assume that a procedure yields a binomial distribution with nequals6 trials and a probability of success of pequals0.40. Use a binomial probability table to find the probability that the number of successes x is exactly 1.
LOADING... Click on the icon to view the binomial probabilities table. 
Question content area bottom
Part 1
Upper P left parenthesis 1 right parenthesisequals
  
enter your response here ​(Round to three decimal places as​ needed.)

In a binomial distribution, the probability of obtaining exactly \( x \) successes in \( n \) trials is given by the formula:

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

Where:
- \( n \) is the number of trials (6 in this case),
- \( x \) is the number of successes (1 in this case),
- \( p \) is the probability of success (0.40 in this case),
- \( \binom{n}{x} \) is the binomial coefficient, which is calculated as \( \frac{n!}{x!(n-x)!} \).

Let's calculate \( P(X = 1) \).

\[
P(X = 1) = \binom{6}{1} (0.40)^1 (1 - 0.40)^{6 - 1}
\]

\[
P(X = 1) = \binom{6}{1} (0.40)(0.60)^5
\]

The binomial coefficient \( \binom{6}{1} \) is 6, so:

\[
P(X = 1) = 6 \times (0.40) \times (0.60)^5
\]

Now, we can compute the probability step by step.

The probability of getting exactly 1 success in 6 trials, with a probability of success of 0.40, is approximately \( P(1) = 0.187 \) (rounded to three decimal places).

Do you want further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the probability change if the number of trials increased to 10?
2. What is the probability of getting exactly 2 successes in 6 trials with the same probability?
3. How do you calculate the cumulative probability for \( X \leq 1 \) in this case?
4. What would happen if the probability of success, \( p \), increased to 0.60?
5. How can you compute the expected value and variance for this binomial distribution?

**Tip:** In binomial distributions, the mean (expected value) is calculated as \( \mu = np \), and the variance as \( \sigma^2 = np(1-p) \).

Assume that a procedure yields a binomial distribution with n=6 trials and a probability of success of p=0.40. Use a binomial probability table to find the probability that the number of successes x is exactly 1. Round to three decimal places as needed.

Learn how to calculate the probability of exactly 1 success in a binomial distribution with 6 trials and a success probability of 0.40. This problem uses the binomial distribution formula and is perfect for high school students studying probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation