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Chapter 4: Problem 12

Defective Parts A spare parts seller finds that 3 in every 100 parts sold is defective. Find the probability that (a) the first defective part is the eighth part sold, (b) the first defective part is the first, second or third part sold, (c) none of the first 8 parts sold are defective.

Short Answer

Expert verified
Probabilities: (a) 0.026, (b) 0.0873, (c) 0.789.

Step by step solution

01

Identify the Probability Parameters

The probability of finding a defective part, as given by the seller, is \( p = \frac{3}{100} = 0.03 \). Therefore, the probability that a part is not defective is \( q = 1 - p = 0.97 \).
02

Solve Part (a)

To find the probability that the first defective part is the eighth part sold, we use the formula for the geometric distribution where the first \( n-1 \) parts are non-defective and the \( n \)-th part is defective. Therefore, for the eighth part: \( P(X = 8) = q^{7} \times p = 0.97^7 \times 0.03 \). Calculating this gives \( P(X = 8) = 0.026 \).
03

Solve Part (b)

Here, we need to find the probability that the first defective part is found within the first three parts sold. This requires summing the probabilities of defectiveness occurring on each of these three trials: \( P(X \leq 3) = (q^{0} \times p) + (q^{1} \times p) + (q^{2} \times p) \). Substituting the values gives: \( 0.03 + 0.97 \times 0.03 + 0.97^2 \times 0.03 = 0.0873 \).
04

Solve Part (c)

For none of the first 8 parts to be defective, all must be non-defective, represented by \( q^8 \). Calculating it gives \( q^8 = 0.97^8 = 0.789 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is a concept that measures how likely an event is to occur. It's expressed as a number between 0 and 1. A probability of 0 means the event will not happen, while a probability of 1 means it will happen for sure. In between, the closer the number is to 1, the higher the likelihood of the event happening.

In this exercise, we consider the probability of a spare part being defective at a rate of 3 out of every 100, which translates to a probability of 0.03. Conversely, the probability of a part not being defective is calculated by subtracting the defective probability from 1. Hence, the probability of a non-defective part is 0.97. This foundation allows us to calculate various scenarios using geometric distribution.

Geometric distribution is particularly useful here as it models the number of trials up to and including the first success (or first defective part in our case). This approach enables us to solve for specific scenarios, such as the first defective coming on a particular sale, or within a set number of sales.
Defective Parts
Defective parts have a specific probability in the context of this problem. A defective part is one that is not functioning as intended or faulty. In the scenario given by the seller, the probability is 3% for each part. Knowing the probability of a defect lets us predict and manage quality control better.

Understanding defective parts probability helps sellers decide on the number of excess parts to keep, anticipating possible defects. This ensures they can meet demand despite defects. Knowing how to calculate scenarios like when the first defect appears can aid in inventory planning and decision-making.

The use of probability in determining defectiveness helps in uncovering patterns over time. If a much higher defect rate is found, for instance, this might prompt a deeper investigation into the manufacturing processes or supplier reliability.
Step-by-step Solution
Breaking down a problem into a step-by-step solution facilitates easier understanding and ensures all aspects are considered. Let's dive deeper into this method using our exercise as the example.

- **Step 1** focuses on identifying the key probability parameters necessary for solving the problem. This includes determining the probability of a defective vs. non-defective part.
- **Step 2** involves applying the geometric distribution formula to calculate when the first defective part is sold. For each part assessed in sequence, the solution involves multiplication of non-defective probabilities followed by one defective probability.
- **Step 3** expands on scenarios where the defective part appears within a specified number of sales. Adding the probabilities for each scenario allows us to comprehend the cumulative probability over those trials.
- **Step 4** is about understanding the scenario where all initial parts are non-defective, using the repeated multiplication of the probability that each part is not defective.
Following a step-by-step method not only helps in accurate calculations but also reinforces understanding, ensuring all calculations are properly accounted for.

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