Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

5. Binomial Distribution & Discrete Random Variables

Hypergeometric Distribution

Statistics

5. Binomial Distribution & Discrete Random Variables

Hypergeometric Distribution

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3:41 minutes

Problem 4.2.41a

Textbook Question

Manufacturing An assembly line produces 10,000 automobile parts. Twenty percent of the parts are defective. An inspector randomly selects 10 of the parts

a. Use the Multiplication Rule (discussed in Section 3.2) to find the probability that none of the selected parts are defective. (Note that the events are dependent.)

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability that none of the 10 randomly selected parts are defective, given that 20% of the 10,000 parts are defective. Since the events are dependent (selecting one part affects the probability of the next), we will use the Multiplication Rule for dependent events.

Step 2: Define the probabilities. The probability of selecting a non-defective part on the first draw is \( P(A_1) = \frac{8000}{10000} \), since 80% of the parts are non-defective. For the second draw, assuming the first part was non-defective, the probability becomes \( P(A_2 | A_1) = \frac{7999}{9999} \), and so on for subsequent draws.

Step 3: Apply the Multiplication Rule for dependent events. The rule states that the probability of all events occurring is the product of their conditional probabilities: \( P(A_1 \cap A_2 \cap \dots \cap A_{10}) = P(A_1) \cdot P(A_2 | A_1) \cdot P(A_3 | A_1 \cap A_2) \cdot \dots \cdot P(A_{10} | A_1 \cap A_2 \cap \dots \cap A_9) \).

Step 4: Write the specific probabilities for each draw. For the 10 draws, the probabilities are: \( P(A_1) = \frac{8000}{10000}, P(A_2 | A_1) = \frac{7999}{9999}, P(A_3 | A_1 \cap A_2) = \frac{7998}{9998}, \dots, P(A_{10} | A_1 \cap A_2 \cap \dots \cap A_9) = \frac{7991}{9991} \).

Step 5: Multiply the probabilities together. To find the overall probability that none of the selected parts are defective, calculate the product: \( P(A_1 \cap A_2 \cap \dots \cap A_{10}) = \frac{8000}{10000} \cdot \frac{7999}{9999} \cdot \frac{7998}{9998} \cdot \dots \cdot \frac{7991}{9991} \). This product gives the desired probability.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Multiplication Rule

The Multiplication Rule in probability is used to find the probability of the occurrence of multiple events. When events are dependent, the probability of the second event is affected by the outcome of the first. This rule states that the probability of two dependent events occurring is the product of the probability of the first event and the conditional probability of the second event given the first.

Dependent Events

Dependent events are those where the outcome of one event affects the outcome of another. In this scenario, selecting a defective part changes the total number of parts available for subsequent selections, thus altering the probabilities. Understanding how these dependencies work is crucial for accurately calculating probabilities in situations like this.

Probability of Defective Parts

In this context, the probability of selecting a defective part is 20%, meaning that out of 10,000 parts, 2,000 are expected to be defective. This probability is essential for calculating the likelihood of selecting non-defective parts. To find the probability that none of the selected parts are defective, one must consider the changing probabilities as parts are selected without replacement.

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