Probability Distribution for Defective Parts

Oct 13, 2025 by ludoplayers7.com 5 views

Introduction

In quality control, it is essential to understand the probability distribution of defective products to ensure that the manufacturing process meets the required standards. Shelia, a quality control specialist, works for a company that produces lawn mower parts. The company has provided her with a probability distribution for defective parts, which is shown in the table below. In this article, we will use the data from the table to calculate the probability of having 2 defective parts in a sample of 5.

Probability Distribution for Defective Parts

Number of Defective Parts	Probability
0	0.2
1	0.3
2	0.2
3	0.2
4	0.1
5	0.1

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

First, we need to calculate the number of combinations of 5 items taken 2 at a time. This can be calculated using the combination formula:

nCk = n! / (k! * (n-k)!)

where n! is the factorial of n.

In this case, we have:

n = 5 k = 2

nCk = 5! / (2! * (5-2)!) = 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 used the data from the table to calculate the probability of having 2 defective parts in a sample of 5. We used the binomial probability formula to calculate the probability, and we found that the probability of having 2 defective parts is approximately 0.1323.

References

Binomial probability formula: P(X = k) = (nCk) * (p^k) * (q^(n-k))
Combination formula: nCk = n! / (k! * (n-k)!)

Introduction

In our previous article, we used the data from the table to calculate the probability of having 2 defective parts in a sample of 5. We used the binomial probability formula to calculate the probability, and we found that the probability of having 2 defective parts is approximately 0.1323. In this article, we will answer some frequently asked questions related to quality control analysis and binomial probability.

Q&A

Q: What is the binomial probability formula?

A: 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)

Q: What is the combination formula?

A: The combination formula is given by:

nCk = n! / (k! * (n-k)!)

where n! is the factorial of n.

Q: How do I calculate the number of combinations?

A: To calculate the number of combinations, you can use the combination formula:

nCk = n! / (k! * (n-k)!)

For example, if you want to calculate the number of combinations of 5 items taken 2 at a time, you would use the following formula:

nCk = 5! / (2! * (5-2)!) = 5! / (2! * 3!) = (5 * 4 * 3 * 2 * 1) / ((2 * 1) * (3 * 2 * 1)) = 10

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, you would use the binomial probability formula:

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, you would use the binomial probability formula:

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, you would use the binomial probability formula:

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 answered some frequently asked questions related to quality control analysis and binomial probability. We used the binomial probability formula to calculate the probability of having 2, 3, 4, and 5 defective parts in a sample of 5. We hope that this article has been helpful in understanding the concept of binomial probability and its application in quality control analysis.

References

Binomial probability formula: P(X = k) = (nCk) * (p^k) * (q^(n-k))
Combination formula: nCk = n! / (k! * (n-k)!)
Quality control: Quality control is the process of ensuring that a product or service meets the required standards. It involves monitoring and controlling the manufacturing process to prevent defects and ensure that the product meets the customer's expectations.