Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

Quality control In a quality control procedure to test for defective light bulbs, two light bulbs are randomly selected from a large sample without replacement. If either light bulb is defective, the entire lot is rejected. Suppose a sample of 200 light bulbs contains 5 defective light bulbs. Find the probability that the sample will be rejected. (Hint: First calculate the probability that neither bulb is defective.)

Step-by-Step Solution

Answer

The probability of the sample being rejected is approximately 0.048.

Step 1: Calculate Probability of First Bulb Being Non-Defective

The probability that the first bulb selected is not defective is calculated by dividing the number of non-defective bulbs by the total number of bulbs.There are 195 non-defective bulbs out of 200, so the probability is \[ P( ext{1st non-defective}) = \frac{195}{200} = 0.975. \]

Step 2: Calculate Probability of Second Bulb Being Non-Defective

If the first bulb is non-defective, there are now 194 non-defective bulbs left out of 199 total bulbs.So, the probability that the second bulb is also non-defective is \[ P( ext{2nd non-defective} | ext{1st non-defective}) = \frac{194}{199}. \]

Step 3: Find Combined Probability of Both Bulbs Non-Defective

To find the probability that both bulbs selected are non-defective, multiply the probabilities from Step 1 and Step 2:\[ P( ext{both non-defective}) = \frac{195}{200} \times \frac{194}{199}. \]

Step 4: Calculate Probability of Sample Being Rejected

The entire lot is rejected if at least one bulb is defective. Therefore, the probability of rejection is the complement of the probability that both bulbs are non-defective. So, \[ P( ext{rejected}) = 1 - P( ext{both non-defective}). \]

Key Concepts

Quality Control

Quality control is an essential part of manufacturing, ensuring that products meet a certain standard before reaching the market. In our exercise, we're looking at a scenario where light bulbs are being tested for defects. The process involves selecting two bulbs at random, and if either is defective, the entire batch from which the bulbs were drawn is rejected.
This scenario highlights the importance of maintaining high quality standards in production.

• Reduced customer dissatisfaction: By catching defective products before they hit the market, companies prevent negative customer experiences.
• Cost efficiency: Identifying defects early can save costs associated with returns and repairs.
• Brand reputation: High quality standards help maintain a company's reputation for reliable products.

Overall, quality control processes are aimed at detecting defects early, and this exercise gives us a probabilistic approach to understanding how that detection is quantified.

Defective Items

Defective items are products that do not meet the required standards of quality and functionality. In our exercise, out of 200 light bulbs, 5 are defective. This illustrates a small defect rate.
Determining the likelihood of picking a defective item helps in making informed decisions about product batches.

• Identifying defects: Spotting defects is crucial for maintaining standards and is why sampling tests, like in our exercise, are valuable.
• Impact on production: A higher rate of defects can mean re-evaluating the production process to increase efficiency and quality.

This concept of defective items emphasizes the importance of statistical insight in quality assurance practices.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arrangement, and selection of objects. In our exercise, it helps us understand the different ways we can select small groups (like the two bulbs) from a larger group (the 200 bulbs).

• Selection: We use combinatorics to calculate the probability of picking items, both defective and non-defective.
• Arrangement: Understanding how items can be selected or arranged helps apply probability rules in quality control.

Combinatorics is crucial in scenarios like ours where the selection probability changes each time an item is sampled. It helps make calculations clear and accurate, aiding in the assessment of quality control outcomes.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. It's a key part of our exercise. After the first bulb is picked and found non-defective, the probabilities for the second bulb change.
Conditional probability helps us update those probabilities in response to new information.

• Initial versus updated probability: Initially, any bulb has a certain chance of being non-defective, but this changes after each selection.
• Understanding interdependent events: Recognizing how events are connected is critical in calculating results accurately in probability theory.

This exercise shows how conditional probability is used in quality control to deal with changing conditions dynamically, ultimately helping determine the likelihood of batch rejection.