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Chapter 4: Problem 11

Find the probability.At Least One. In Exercises \(5-12,\) find the probability. It has been reported that \(20 \%\) of iPhones manufactured by Foxconn for a product launch did not meet Apple's quality standards. An engineer needs at least one defective iPhone so she can try to identify the problem(s). If she randomly selects 15 iPhones from a very large batch, what is the probability that she will get at least 1 that is defective? Is that probability high enough so that she can be reasonably sure of getting a defect for her work?

Short Answer

Expert verified
The probability is approximately 0.9648, or 96.48%, which is high enough to be reasonably sure of getting a defective iPhone for the engineer's work.

Step by step solution

01

Understand the Problem

We need to find the probability that at least one of the selected iPhones will be defective. Given data: The probability that an iPhone is defective is 0.20. We are selecting 15 iPhones randomly.
02

Find the Probability of No Defective iPhone

First, calculate the probability that none of the 15 selected iPhones is defective. The probability that an iPhone is not defective is 0.80. Using this, the probability that all 15 iPhones are not defective is The probability that all 15 iPhones are not defective is: 0.80^15.
03

Calculate the Probability of No Defective iPhones

Calculate the value for 0.80^15. The numerical result is approximately 0.0352.
04

Find the Probability of At Least One Defective iPhone

The probability of getting at least one defective iPhone is the complement of the probability of getting no defective iPhones. This is calculated as 1 minus the probability of no defective iPhones. So, 1 - 0.0352 which equals 0.9648.

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Key Concepts

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

Probability of at least one defective iPhone
To find the probability of getting at least one defective iPhone, we need to understand that 'at least one' is a key probability concept. It means we are looking for one or more defective iPhones among our selection.
This concept involves looking not just at exact counts, but any scenario where at least one is found. By focusing on 'at least one,' we cover all possible cases except when there are none.
Using the binomial distribution
The binomial distribution helps us find probabilities when we have a fixed number of independent trials, like checking 15 iPhones. Each trial has two outcomes: success (defective) and failure (not defective).
The parameters we need are the number of trials n (15 iPhones) and the probability of success p (0.20 for defective). To find the probability of specific outcomes, we also use the binomial formula. However, for finding 'at least one,' focusing on the complement simplifies our calculations. See how we examine the 'no defective Phone' case first.
Understanding the complement rule
The complement rule is a powerful tool in probability. It states that the probability of an event happening is 1 minus the probability of it not happening. This is written as: \( P(A) = 1 - P(A^c) \)
Applying this to our problem, we can avoid calculating each scenario where there is 1, 2, or more defective iPhones by instead finding the probability of having zero defective iPhones and subtracting that from 1. This avoids complex calculations and makes our problem much simpler.
Probability of defective items
For our exercise, \( P(\text{def} = \text{defective}) = 0.20 \), which means the probability of getting a defective iPhone is 20%.
The probability of getting a non-defective iPhone is therefore \( P(\text{not def}) = 0.80 \).
Given this, for 15 iPhones, the probability that none are defective (all are non-defective) is \( 0.80^{15} \approx 0.0352 \). Thus, using our complement rule, the probability of getting at least one defective iPhone is: \( 1 - 0.0352 \approx 0.9648 \).
This high probability indicates a strong chance of getting what the engineer needs for her inspection.

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