Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

Problem 4.2.41b

Chapter 4, Problem 4.2.41b

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

b. Because the sample is only 0.1% of the population, treat the events as independent and use the binomial probability formula to approximate the probability that none of the selected parts are defective.

Verified step by step guidance

Step 1: Identify the parameters of the binomial distribution. The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success (p). Here, n = 10 (the number of parts selected) and p = 0.2 (the probability that a part is defective). The probability of a part not being defective is 1 - p = 0.8.

Step 2: Write the binomial probability formula. The formula for the probability of exactly k successes in n trials is: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k), where (n choose k) = n! / [k! * (n - k)!].

Step 3: Substitute the values into the formula. Since we are calculating the probability that none of the selected parts are defective (k = 0), the formula becomes: P(X = 0) = (10 choose 0) * (0.2)^0 * (0.8)^10.

Step 4: Simplify the formula. (10 choose 0) = 1 because there is only one way to choose 0 items from 10. Also, (0.2)^0 = 1. Therefore, the formula simplifies to: P(X = 0) = 1 * 1 * (0.8)^10.

Step 5: Calculate (0.8)^10. This involves raising 0.8 to the power of 10. The result will give you the probability that none of the selected parts are defective.

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Key Concepts

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

Binomial Probability Distribution

The binomial probability distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, a 'success' refers to selecting a defective part. The formula for calculating the probability of exactly k successes in n trials is given by P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success.

Independence of Events

In probability theory, two events are considered independent if the occurrence of one does not affect the probability of the other occurring. In this scenario, treating the selection of parts as independent is justified because the sample size (10 parts) is small relative to the total population (10,000 parts), allowing us to ignore the slight changes in probabilities that would occur if the selections were dependent.

Probability of No Defective Parts

To find the probability that none of the selected parts are defective, we can use the binomial formula with k = 0. This means we are interested in the probability of selecting 0 defective parts out of 10, given that the probability of selecting a defective part is 20% (or 0.2). The calculation simplifies to P(X = 0) = (1 - p)^n, which in this case is (0.8)^10, representing the probability that all selected parts are non-defective.

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