How to Calculate the Probability of Detecting Defective Lightbulbs During Manufacturing?
Producing lightbulbs isn’t just a technically demanding and painstaking process; it’s also an art that necessitates constant quality control at every stage. Despite all efforts and high manufacturing standards, there’s always a chance that defective products will slip through. But how can we calculate the odds that, out of three randomly selected lightbulbs, all three will be defective? Let’s dive into this question and find out.
Let’s assume the defect rate in lightbulb manufacturing is 10%, and suppose 1,000 lightbulbs are produced. This means 100 out of the 1,000 lightbulbs will be defective.
Here are a few illustrative examples:
- If 7,000 lightbulbs are produced in a week, and the defect rate remains unchanged, around 700 defective lightbulbs can be expected.
- Similarly, at another factory producing 500 lightbulbs daily, with the same defect rate, about 50 defective lightbulbs would be deemed unacceptable each day.
Returning to our initial question, the probability of selecting one defective lightbulb is 0.1 (or 10%). This is calculated by dividing the number of defective lightbulbs (100) by the total number of lightbulbs (1,000).
Now, to find the probability that all three chosen lightbulbs will be defective, we need to multiply the probability for each one. Since each selection is an independent event, we use the formula P(A∩B∩C) = P(A) * P(B) * P(C), where P(A), P(B), and P(C) are the probabilities of selecting a defective lightbulb on the first, second, and third tries, respectively.
So, P(A) = 0.1, P(B) = 0.1, and P(C) = 0.1. Multiplying these probabilities gives:
0.1 * 0.1 * 0.1 = 0.001
Thus, the probability that all three randomly selected lightbulbs will be defective is 0.001, or one in a thousand. This is quite an unlikely scenario, underscoring the importance of rigorous quality control in manufacturing to minimize the risk of defective products.
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