Shelia Does Quality Control For A Company That Manufactures Lawn Mower Parts. On Any Given Day, She Finds The Probability Distribution For Defective Parts As Shown In The Table.1. Using The Data From The Table, What Is The Probability Of Having 2
Introduction
In the manufacturing industry, quality control is a crucial aspect that ensures the production of high-quality products. One of the key components of quality control is the analysis of probability distributions, which helps in understanding the likelihood of defects in products. In this article, we will discuss the probability distribution of defective lawn mower parts, as observed by Shelia, a quality control specialist. We will use the data from the table to calculate the probability of having 2 defective parts in a sample of 5.
Understanding the Probability Distribution
The probability distribution of defective lawn mower parts is shown in the table below:
| Number of Defects | Probability |
|---|---|
| 0 | 0.2 |
| 1 | 0.3 |
| 2 | 0.2 |
| 3 | 0.15 |
| 4 | 0.1 |
| 5 | 0.05 |
Calculating the Probability of Having 2 Defective Parts
To calculate the probability of having 2 defective parts in a sample of 5, we need to use the binomial probability formula. The binomial probability formula is given by:
P(X = k) = (nCk) * (p^k) * (q^(n-k))
where:
- P(X = k) is the probability of having k defective parts
- n is the sample size (in this case, 5)
- k is the number of defective parts (in this case, 2)
- nCk is the number of combinations of n items taken k at a time
- p is the probability of a part being defective (in this case, 0.2 + 0.3 + 0.2 = 0.7)
- q is the probability of a part not being defective (in this case, 1 - p = 0.3)
Step 1: Calculate the Number of Combinations
To calculate the number of combinations of 5 items taken 2 at a time, we use the formula:
nCk = n! / (k! * (n-k)!)
where:
- n! is the factorial of n (in this case, 5!)
- k! is the factorial of k (in this case, 2!)
- (n-k)! is the factorial of (n-k) (in this case, 3!)
5C2 = 5! / (2! * 3!) = (5 * 4 * 3 * 2 * 1) / ((2 * 1) * (3 * 2 * 1)) = 10
Step 2: Calculate the Probability of Having 2 Defective Parts
Now that we have the number of combinations, we can calculate the probability of having 2 defective parts using the binomial probability formula:
P(X = 2) = (nCk) * (p^k) * (q^(n-k)) = 10 * (0.7^2) * (0.3^3) = 10 * 0.49 * 0.027 = 0.1323
Conclusion
In this article, we analyzed the probability distribution of defective lawn mower parts, as observed by Shelia, a quality control specialist. We used the data from the table to calculate the probability of having 2 defective parts in a sample of 5. We found that the probability of having 2 defective parts is approximately 0.1323. This information can be used by quality control specialists to improve the manufacturing process and reduce the likelihood of defects in products.
References
- [1] Shelia's Quality Control Data (Table)
- [2] Binomial Probability Formula
Additional Resources
- [1] Quality Control: A Guide to Understanding Probability Distributions
- [2] Binomial Probability Calculator
Frequently Asked Questions
- Q: What is the probability of having 2 defective parts in a sample of 5? A: The probability of having 2 defective parts in a sample of 5 is approximately 0.1323.
- Q: How can I improve the manufacturing process to reduce the likelihood of defects?
A: You can use the information from this article to improve the manufacturing process by reducing the probability of defects in products.
Quality Control Analysis: Understanding the Probability Distribution of Defective Lawn Mower Parts ===========================================================
Q&A: Frequently Asked Questions
Q: What is the probability of having 0 defective parts in a sample of 5? A: To calculate the probability of having 0 defective parts in a sample of 5, we need to use the binomial probability formula. The probability of a part not being defective is 0.3, and the number of combinations of 5 items taken 0 at a time is 1. Therefore, the probability of having 0 defective parts is:
P(X = 0) = (nCk) * (p^k) * (q^(n-k)) = 1 * (0.7^0) * (0.3^5) = 1 * 1 * 0.243 = 0.243
Q: What is the probability of having 1 defective part in a sample of 5? A: To calculate the probability of having 1 defective part in a sample of 5, we need to use the binomial probability formula. The probability of a part being defective is 0.7, and the number of combinations of 5 items taken 1 at a time is 5. Therefore, the probability of having 1 defective part is:
P(X = 1) = (nCk) * (p^k) * (q^(n-k)) = 5 * (0.7^1) * (0.3^4) = 5 * 0.49 * 0.082 = 0.201
Q: What is the probability of having 3 defective parts in a sample of 5? A: To calculate the probability of having 3 defective parts in a sample of 5, we need to use the binomial probability formula. The probability of a part being defective is 0.7, and the number of combinations of 5 items taken 3 at a time is 10. Therefore, the probability of having 3 defective parts is:
P(X = 3) = (nCk) * (p^k) * (q^(n-k)) = 10 * (0.7^3) * (0.3^2) = 10 * 0.343 * 0.09 = 0.3087
Q: What is the probability of having 4 defective parts in a sample of 5? A: To calculate the probability of having 4 defective parts in a sample of 5, we need to use the binomial probability formula. The probability of a part being defective is 0.7, and the number of combinations of 5 items taken 4 at a time is 5. Therefore, the probability of having 4 defective parts is:
P(X = 4) = (nCk) * (p^k) * (q^(n-k)) = 5 * (0.7^4) * (0.3^1) = 5 * 0.2401 * 0.3 = 0.36015
Q: What is the probability of having 5 defective parts in a sample of 5? A: To calculate the probability of having 5 defective parts in a sample of 5, we need to use the binomial probability formula. The probability of a part being defective is 0.7, and the number of combinations of 5 items taken 5 at a time is 1. Therefore, the probability of having 5 defective parts is:
P(X = 5) = (nCk) * (p^k) * (q^(n-k)) = 1 * (0.7^5) * (0.3^0) = 1 * 0.16807 * 1 = 0.16807
Conclusion
In this article, we provided answers to frequently asked questions related to the probability distribution of defective lawn mower parts. We used the binomial probability formula to calculate the probabilities of having 0, 1, 2, 3, 4, and 5 defective parts in a sample of 5. We hope that this information will be helpful to quality control specialists and manufacturers who want to improve the quality of their products.
References
- [1] Shelia's Quality Control Data (Table)
- [2] Binomial Probability Formula
Additional Resources
- [1] Quality Control: A Guide to Understanding Probability Distributions
- [2] Binomial Probability Calculator