Calculating the Probability of Defective Car Brakes in a Batch
A manufacturer of car brakes finds that on average, 5% of the car brakes produced are rejected because they are either oversize or undersize. This article explores the probability that a batch of 8 car brakes will contain no more than 2 rejects, using statistical methods and binomial distribution principles.
Statistical Analysis
Let's break down the calculations to determine the probability of having 0, 1, or 2 defective brakes in a batch of 8.
No Rejections (0 Rejects)
The probability of having no rejects in a batch of 8 brakes is calculated as follows:
Probability of no rejects 0.95^8 ≈ 0.6634
One Rejection (1 Reject)
The probability of having exactly one reject is given by the binomial coefficient C(8, 1), which represents the number of ways to choose 1 reject out of 8 brakes, multiplied by the probability of having one reject and seven non-rejects:
Probability of one reject C(8, 1) * (0.05)^1 * (0.95)^7 ≈ 0.2793
Two Rejections (2 Rejects)
The probability of having exactly two rejects is calculated using the binomial coefficient C(8, 2), which represents the number of ways to choose 2 rejects out of 8 brakes, multiplied by the probability of having two rejects and six non-rejects:
Probability of two rejects C(8, 2) * (0.05)^2 * (0.95)^6 ≈ 0.0515
Summarizing Probabilities
Adding these probabilities together, we find the total probability of having no more than 2 rejects:
Total probability Probability of 0 rejects Probability of 1 reject Probability of 2 rejects ≈ 0.6634 0.2793 0.0515 ≈ 0.9942
Application of Binomial Distribution
The method used to calculate the probabilities in this scenario is based on the binomial distribution. The formula for the binomial probability is:
P(n) (8 C n) * (0.05)^n * (0.95)^(8-n)
Where:
P(n) is the probability of having exactly n defective brakes. (8 C n) is the binomial coefficient representing the number of ways to choose n defects out of 8 tries. 0.05 is the probability of a single brake being defective. 0.95 is the probability of a single brake not being defective.Real-World Significance
Understanding these probabilities is crucial for a manufacturer to meet quality control standards and ensure customer satisfaction. The statistical analysis indicates that 99.42% of the time, a batch of 8 car brakes will have no more than 2 rejects. This is a valuable insight for both the manufacturing process and quality assurance practices.
Conclusion
The use of binomial distribution in this scenario provides a clear and systematic way to calculate the probabilities of defective components in a batch. This method is widely applicable in manufacturing and quality control, helping to ensure that products meet high standards of quality and reliability.