Problem 34 - Algebra and Trigonometry

Question

Quality Control To control the quality of their product, the Bright-Light Company inspects three light bulbs out of each batch of ten bulbs manufactured. If a defective bulb is found, the batch is discarded. Suppose a batch contains two defective bulbs. What is the probability that the batch will be discarded?

Step-by-Step Solution

Answer

The probability that the batch will be discarded is \( \frac{8}{15} \).

Step 1: Understand the Problem

We are given a batch of 10 bulbs containing 2 defective bulbs. We need to find the probability that in a random selection of 3 bulbs, at least one is defective.

Step 2: Define Possible Outcomes

We can calculate probabilities by defining different scenarios when picking 3 bulbs out of 10. We are looking for scenarios where at least one defective bulb is picked. Consequently, it is easier to compute the complementary probability where no defective bulbs are selected and subtract it from 1.

Step 3: Compute Total Ways to Select 3 Bulbs

The total number of ways to choose 3 bulbs from 10 is given by the combination formula \( \binom{10}{3} = \frac{10!}{3!(10-3)!} = 120 \).

Step 4: Compute Ways to Select 3 Non-defective Bulbs

Since there are 8 non-defective bulbs, the number of ways to choose 3 non-defective bulbs is \( \binom{8}{3} = \frac{8!}{3!(8-3)!} = 56 \).

Step 5: Calculate the Complement Probability

The probability of selecting only non-defective bulbs is the number of ways to select 3 non-defective bulbs divided by the total ways to choose 3 bulbs: \( \frac{56}{120} = \frac{7}{15} \).

Step 6: Find the Probability of Finding At Least One Defective Bulb

The probability that at least one defective bulb is found is the complement of selecting only non-defective bulbs: \( 1 - \frac{7}{15} = \frac{8}{15} \).

Key Concepts

Understanding Quality Control

Quality control is a crucial aspect in the manufacturing process. It ensures that products meet a certain standard before they reach consumers. This process often involves inspecting a random sample from each batch of products. For instance, if a company produces light bulbs, they will inspect a small number, like three out of ten, from each batch. This inspection allows the company to identify and eliminate faulty batches before they are shipped out. Maintaining quality control is vital because it:

• Helps maintain customer satisfaction and trust.
• Reduces the risk of product recalls, saving costs in the long run.
• Ensures compliance with industry standards and safety regulations.

Through effective quality control, companies can minimize the number of defective products reaching the consumer.

Identifying Defective Products

Defective products are items that fail to meet the designed specifications or consumer expectations. These defects could arise due to errors in material, production, or design. In our example, a batch of light bulbs is said to have two defective bulbs. Such defective products can:

• Lead to customer dissatisfaction and potential harm.
• Create financial loss due to returns and recalls.
• Damage the company's reputation if not managed properly.

Identifying defects early in the production process is key. Companies aim to detect these issues by statistically analyzing batches of products. Sampling techniques allow firms to inspect a small subset of the batch, which represents the entire lot. If defects are discovered consistently in samples, the entire batch may be rejected, ensuring that only quality items proceed to market.

The Role of Combinatorics in Probability Analysis

Combinatorics is a branch of mathematics dealing with the counting, arrangement, and combination of objects. It plays a critical role in probability theory by helping to calculate the likelihood of specific outcomes, especially in quality control problems.In our light bulb example, combinatorics determines how to count the different ways to select samples from a batch. Let's break it down:

• Total Combinations: To find how to pick 3 bulbs from 10, we use the combination formula \( \binom{10}{3} \), which calculates to 120 ways.
• Non-defective Selections: Since there are 8 non-defective bulbs, selecting 3 from 8 involves \( \binom{8}{3} \), giving 56 ways.
• Probability Analysis: By dividing the number of non-defective selections by the total combinations (\( \frac{56}{120} \)), we determine the probability of picking only non-defective bulbs.

This mathematical foundation allows us to assess the probability of drawing at least one defective bulb, which is crucial for quality control decisions. Combinatorics thus provides a systematic method for evaluating possible outcomes and making informed decisions based on statistical evidence.