Understanding Probability in Quality Engineering: A Focus on Nonconforming Pieces

What is the probability that nonconforming pieces will not be found in a sample of 50 if the process produces 0.2% nonconforming pieces?

• A. 0.09
• B. 0.36
• C. 0.64
• D. 0.91

Answer: (D)

To determine the probability that nonconforming pieces will not be found in a sample of 50 when the process produces a rate of 0.2% nonconforming pieces, we first express the nonconforming rate as a decimal: 0.2% is equivalent to 0.002. This means that the probability of a piece being conforming is 1 - 0.002, which equals 0.998. Since we want to find out the probability that all 50 pieces in the sample are conforming, we can use the following formula for independent events: Probability(all conforming) = (Probability of a conforming piece) ^ (number of pieces in the sample) Thus, we calculate: Probability(all conforming) = 0.998^50 When you calculate this: 0.998^50 ≈ 0.91 This shows a strong probability that no nonconforming pieces will be present in the sample of 50. Therefore, option D, which brings this probability into the context of 0.91, accurately represents the situation where, despite a small percentage of nonconforming pieces, the likelihood of selecting a sample of 50 and finding none is quite high.

Ever found yourself scratching your head over a probability question? You’re not alone! Let’s break down a classic problem relevant to quality engineering—specifically, the probability of nonconforming pieces. Imagine this: a process that churns out goods has a tiny nonconformance rate of just 0.2%. That means, on average, only two out of every thousand pieces are not up to par. So, what does that mean when you take a sample—a batch of 50 pieces?

First, let’s consider how we express the nonconformance rate in a way that’s useful for calculations. We know that 0.2% translates to 0.002 in its decimal form. This brings us to a more favorable number for calculations—the probability of a piece being conforming: 1 minus the nonconformance rate. So, that’s 1 - 0.002 = 0.998.

Now, here’s where it gets interesting. We want to find out whether we’ll get an entire sample of 50 pieces without encountering any nonconforming ones. It’s like checking a batch of cookies to see if any are burnt—you really don’t want that! For independent events (think of each cookie baked), the probability of all pieces being conforming can be calculated with a neat little formula:

Probability(all conforming) = (Probability of a conforming piece) ^ (number of pieces).

Plugging in our numbers gives us:

Probability(all conforming) = 0.998^50.

When we crunch those numbers—drum roll, please—we find:

0.998^50 is about 0.91.

This tells us that despite the low nonconformance rate, there’s a solid 91% chance that in our sample of 50 pieces, all will be conforming. Imagine if you were in a quality assurance role and this came up—knowing that there's a high probability of getting a perfect batch can make a huge difference in your planning.

So, when you’re preparing for your Certified Quality Engineer exam, keep these calculations in your toolkit. Understanding how to navigate probabilities—like anticipating nonconformance—is an essential skill in maintaining quality assurance. And don’t forget, while the math is vital, the real-world application will be what sets you apart, so make sure you practice applying this knowledge in various scenarios.