A binomial experiment with probability of success p=0.12 and n=6 trials is conducted. What is the probability that the experiment results in more than 2 successes?
A binomial experiment with probability of success p=0.12 and n=6 trials is conducted. What is the probability that the experiment results in more than 2 successes?
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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![### Understanding Binomial Experiments: Calculating the Probability of Success
**Scenario:**
A binomial experiment is conducted with the probability (p) of success being 0.12, and the number of trials (n) set to 6. We aim to determine the probability that this experiment yields more than 2 successes.
**Procedure:**
To solve this, we need to calculate the probability of getting more than 2 successes in 6 trials. Specifically, we need to find:
\[ P(X > 2) \]
Where \(X\) is the random variable representing the number of successes in the trials.
**Steps:**
1. First, note that the binomial probability formula is given by:
\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \]
Here, \(\binom{n}{k}\) is the binomial coefficient, which calculates the number of ways to choose \(k\) successes out of \(n\) trials.
2. For our problem, we should find the cumulative probability for 0 to 2 successes and subtract this from 1 to find the probability of more than 2 successes.
3. Calculate probabilities for \(X = 0\), \(X = 1\), and \(X = 2\):
- For \( X = 0 \):
\[
P(X = 0) = \binom{6}{0} \cdot (0.12)^0 \cdot (0.88)^6
\]
- For \( X = 1 \):
\[
P(X = 1) = \binom{6}{1} \cdot (0.12)^1 \cdot (0.88)^5
\]
- For \( X = 2 \):
\[
P(X = 2) = \binom{6}{2} \cdot (0.12)^2 \cdot (0.88)^4
\]
4. Sum the probabilities for 0 to 2 successes:
\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]
5. Finally, subtract the cumulative probability from](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff55e73a7-a345-453e-a44c-cf1798955df8%2F063a5430-8604-4941-8b8d-e62fbb4193c9%2Fekdwcz8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Binomial Experiments: Calculating the Probability of Success
**Scenario:**
A binomial experiment is conducted with the probability (p) of success being 0.12, and the number of trials (n) set to 6. We aim to determine the probability that this experiment yields more than 2 successes.
**Procedure:**
To solve this, we need to calculate the probability of getting more than 2 successes in 6 trials. Specifically, we need to find:
\[ P(X > 2) \]
Where \(X\) is the random variable representing the number of successes in the trials.
**Steps:**
1. First, note that the binomial probability formula is given by:
\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \]
Here, \(\binom{n}{k}\) is the binomial coefficient, which calculates the number of ways to choose \(k\) successes out of \(n\) trials.
2. For our problem, we should find the cumulative probability for 0 to 2 successes and subtract this from 1 to find the probability of more than 2 successes.
3. Calculate probabilities for \(X = 0\), \(X = 1\), and \(X = 2\):
- For \( X = 0 \):
\[
P(X = 0) = \binom{6}{0} \cdot (0.12)^0 \cdot (0.88)^6
\]
- For \( X = 1 \):
\[
P(X = 1) = \binom{6}{1} \cdot (0.12)^1 \cdot (0.88)^5
\]
- For \( X = 2 \):
\[
P(X = 2) = \binom{6}{2} \cdot (0.12)^2 \cdot (0.88)^4
\]
4. Sum the probabilities for 0 to 2 successes:
\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]
5. Finally, subtract the cumulative probability from
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