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.)

Accepted Answer

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

Multiplication Rule

Dependent Events

Probability of Defective Parts