A binomial experiment with probability of success p=0.4 and n=5 trials is conducted. What is the probability that the experiment results in exactly 1 success? Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.) 0 S ?
A binomial experiment with probability of success p=0.4 and n=5 trials is conducted. What is the probability that the experiment results in exactly 1 success? Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.) 0 S ?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Binomial Probability Calculation
**Problem Statement:**
A binomial experiment with a probability of success \( p = 0.4 \) and \( n = 5 \) trials is conducted. What is the probability that the experiment results in exactly 1 success?
Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)
---
### Explanation
In binomial experiments, the probability of exactly \( k \) successes in \( n \) trials is given by:
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} \]
Where:
- \( n \) is the number of trials
- \( k \) is the number of successes
- \( p \) is the probability of success on a single trial
- \( \binom{n}{k} \) is the binomial coefficient, calculated as:
\[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \]
Plugging in the values from the given problem:
- \( n = 5 \)
- \( k = 1 \)
- \( p = 0.4 \)
- \( (1 - p) = 0.6 \)
So,
\[ P(X = 1) = \binom{5}{1} (0.4)^1 (0.6)^{5-1} \]
\[ P(X = 1) = 5 \cdot 0.4 \cdot (0.6)^4 \]
\[ P(X = 1) = 5 \cdot 0.4 \cdot 0.1296 \]
\[ P(X = 1) = 5 \cdot 0.05184 \]
\[ P(X = 1) = 0.2592 \]
After rounding to three decimal places, the probability that the experiment results in exactly 1 success is:
\[ \boxed{0.259} \]
Note: There are no graphs or diagrams included in this specific problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff55e73a7-a345-453e-a44c-cf1798955df8%2F99a713e2-2e16-4b12-ad79-f5dfc02c22fe%2Fli9e9g3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Binomial Probability Calculation
**Problem Statement:**
A binomial experiment with a probability of success \( p = 0.4 \) and \( n = 5 \) trials is conducted. What is the probability that the experiment results in exactly 1 success?
Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)
---
### Explanation
In binomial experiments, the probability of exactly \( k \) successes in \( n \) trials is given by:
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} \]
Where:
- \( n \) is the number of trials
- \( k \) is the number of successes
- \( p \) is the probability of success on a single trial
- \( \binom{n}{k} \) is the binomial coefficient, calculated as:
\[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \]
Plugging in the values from the given problem:
- \( n = 5 \)
- \( k = 1 \)
- \( p = 0.4 \)
- \( (1 - p) = 0.6 \)
So,
\[ P(X = 1) = \binom{5}{1} (0.4)^1 (0.6)^{5-1} \]
\[ P(X = 1) = 5 \cdot 0.4 \cdot (0.6)^4 \]
\[ P(X = 1) = 5 \cdot 0.4 \cdot 0.1296 \]
\[ P(X = 1) = 5 \cdot 0.05184 \]
\[ P(X = 1) = 0.2592 \]
After rounding to three decimal places, the probability that the experiment results in exactly 1 success is:
\[ \boxed{0.259} \]
Note: There are no graphs or diagrams included in this specific problem.
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