Probability Distribution: Analyzing the Defective Rate in LED Bulb Production
In the manufacturing process of LED bulbs, it is crucial for manufacturers to understand the probability distribution of defective bulbs in order to ensure product quality and customer satisfaction. A common scenario involves knowing the expected number of defective bulbs in a carton of 144 bulbs, given that 3 out of every 100 bulbs produced are defective.
Understanding the Probability
Given that the probability of a bulb being defective, P(D), is 4 out of 100, or 0.04, we can calculate the probability that any bulb being non-defective as P(N) 1 - P(D) 0.96. For a carton of 144 bulbs, the probability of exactly 6 bulbs being defective can be calculated using the binomial probability formula:
Using the Binomial Probability Formula
The formula for the probability of exactly k defective bulbs in n trials is given by:
P(k bad) nk pk (1 - p)n-k
Where:
n is the total number of bulbs, which is 144. k is the number of defective bulbs, which is 6. q is the probability of a bulb not being defective, which is 0.96.Substituting the values, we get:
P(6 bad) 142C6 0.046 0.96142-6
The binomial coefficient 142C6 is calculated as follows:
142C6 142! / (6! times; 136!)
Calculating this, we get the probability as:
P(6 bad) 10230985051 times; 0.046 times; 0.96138 ≈ 0.16261668
Interpreting the Results
This means that there is approximately a 16.26% chance that exactly 6 out of 144 bulbs will be defective. Additionally, the probability distribution shows that the chances of having between 3 and 7 defective bulbs are over 70%, indicating a reasonable probability for the expected value of 5.68 defective bulbs.
Practical Implications
Understanding this probability distribution is crucial for making informed decisions in quality control and manufacturing processes. It helps manufacturers and quality assurance teams to:
Identify potential issues with production lines and machinery. Adjust production processes to reduce the rate of defective products. Ensure customer satisfaction by guaranteeing product quality.Conclusion
While the probability of having exactly 6 defective bulbs in a carton of 144 is significant, it's essential to recognize that the actual defect rates can vary. This probability distribution provides a valuable tool for manufacturers to monitor and improve their production processes. Knowledge of such probabilities helps in making data-driven decisions that ultimately contribute to higher product quality and customer satisfaction.
Related Questions and Further Reading
For further discussion, consider the following questions and resources:
How do you arrange 142 bulbs in a carton? This involves space optimization and practical packaging. Why are LED bulbs more efficient? Explore the advancements in LED technology and energy savings compared to incandescent bulbs. Probability and Statistical Distributions - Dive deeper into statistical methods and their applications in industry.