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Chapter 15: Problem 20

Suppose a computer chip manufacturer rejects \(2 \%\) of the chips produced because they fail presale testing. a. What's the probability that the fifth chip you test is the first bad one you find? b. What's the probability you find a bad one within the first 10 you examine?

Short Answer

Expert verified
a) The probability that the fifth chip you test is the first bad one is calculated by \(P(X=5) = (1 - 0.02)^{5-1} * 0.02\). b) The probability that a defective chip is found within the first ten chips can be found by calculating \(P(X>10) = (1 - 0.02)^{10}\), then subtract that value from 1, indicating the complement of the event, \(P(X<=10) = 1 - P(X>10)\). In both calculations, perform the necessary computations to get the final probability values.

Step by step solution

01

- Calculate the probability of first defect on fifth test

In order to find the probability that the first defective chip is found on the fifth test, a geometric distribution model can be applied. The corresponding formula for the geometric distribution is given by\(P(X=k) = (1-p)^{k-1} * p\)where \(p\) is the 'success' probability, \((1-p)\) is the 'failure' probability, and \(k\) is the number of trials until the first success. So for this part of the exercise, \(p\) is calculated as 0.02 (representing the 2% defect probability), and \(k\) is 5, since we are looking for the first defect on the fifth test.Substitute the given values into the formula:\(P(X=5) = (1 - 0.02)^{5-1} * 0.02\)
02

- Calculate the probability of finding a defect within the first 10 tests

For the second part, we need to find the probability that at least one defective chip is found within the first ten tests. This is a form of negative binomial distribution problem, where the target is not the nth success, but rather the first success. However, this can be approached more intuitively by calculating the probability of not finding any defective chips within the first ten tests and subtracting it from 1. This is because the sum of probabilities of all possible events is equal to 1.The formula for the probability for \(k\) failures before the first success in a geometric distribution is\(P(X>k) = (1-p)^k\)In our case, \(k\) is 10 and \(p\) is 0.02. Substituting the given values:\(P(X>10) = (1 - 0.02)^{10}\)In order to find the probability that at least one defective chip is found in the first 10 chips, subtract the calculated value from 1.
03

- Perform Calculations

Carry out the mathematical calculations found in Step 1 and Step 2, which will provide the final probability values to be reported. Perform the calculations using a calculator or software equipped to handle exponential computations.

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Key Concepts

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

Probability
Understanding probability is fundamental in statistics when predicting the likelihood of different outcomes. It is essentially the measure of how likely an event is to occur. The probability value ranges between 0 and 1, where 0 indicates impossibility, and 1 suggests certainty.

In practical terms, when we say the probability of an event is 0.5, we mean there's a 50% chance of it occurring. Probability takes many forms depending on the context. For example, the rolling of a die involves basic probability, while more complex forms like **geometric** or **negative binomial distribution** apply to series of dependent events.
  • Independent events: The outcome of one event does not affect the other.
  • Dependent events: The outcome of one event influences the probability of another event.
Recognizing the type of event is crucial for choosing the correct probability model.
Defective Chips
The scenario in the exercise revolves around defective computer chips. Defective chips are those that fail the quality control and cannot be sold.

This situation mirrors many industrial processes where a small percentage of products are expected not to meet the quality standards. For this exercise, 2% are expected to be defective, hence considered failures during quality checks.

Managing defective items is crucial in manufacturing as it impacts both profitability and customer satisfaction. Detecting these defective chips early鈥攂efore they reach consumers鈥攊s essential for maintaining quality assurance. This exercise uses statistical methods to determine how likely it is to encounter a defective chip during the sampling process.
Negative Binomial Distribution
The negative binomial distribution is a probability distribution that deals with a sequence of independent and identically distributed Bernoulli trials.

Unlike simple Bernoulli or binomial distributions, which predict outcomes over fixed numbers of trials, negative binomial distribution focuses on the number of trials needed to achieve a fixed number of successful outcomes.

This is especially useful when you're looking for the occurrence of the first success or failure after a certain number of trials. It's applicable in situations such as quality control in manufacturing, as seen in this exercise about defective chips.
  • In the context: The **negative binomial** is applied to measure occurrences before the first defective chip is found.
  • This approach is used to find probabilities over repeated trials until achieving the first failure.
Negative binomial calculations can often be simplified by referencing cumulative probabilities, optimizing the process.
Geometric Probability Formula
The geometric probability formula is pivotal in scenarios where events repeat until the first success, like in the search for the first defective chip.

The formula itself is:\[P(X=k) = (1-p)^{k-1} \cdot p\]Where:
  • **\(P(X=k)\):** Probability that the first success occurs on the kth trial.
  • **\(p\):** Probability of success on an individual trial.
  • **\((1-p)\):** Probability of failure on a single trial.
  • **\(k\):** Trial number on which first success is achieved.
In this exercise, using the geometric probability formula, we can compute the chance that the first bad chip shows up on the fifth test. This highlights the practical application of the formula in real-world quality testing scenarios.

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Most popular questions from this chapter

You are one space short of winning a child's board game and must roll a 1 on a die to claim victory. You want to know how many rolls it might take. a. Describe how you would simulate rolling the die until you get a 1 . b. Run at least 30 trials. c. Based on your simulation, estimate the probabilities that you might win on the first roll, the second, the third, etc. d. Calculate the actual probability model. e. Compare the distribution of outcomes in your simulation to the probability model.

A basketball player has made \(80 \%\) of his foul shots during the season. Assuming the shots are independent, find the probability that in tonight's game he a. misses for the first time on his fifth attempt. b. makes his first basket on his fourth shot. c. makes his first basket on one of his first 3 shots.

A Department of Transportation report about air travel found that airlines misplace about 5 bags per 1000 passengers. Suppose you are traveling with a group of people who have checked 22 pieces of luggage on your flight. Can you consider the fate of these bags to be Bernoulli trials? Explain.

A lecture hall has 200 seats with folding arm tablets, 30 of which are designed for left-handers. The average size of classes that meet there is \(188,\) and we can assume that about \(13 \%\) of students are left-handed. What's the probability that a right-handed student in one of these classes is forced to use a lefty arm tablet?

The basketball player in Exercise 55 has new sneakers, which he thinks improve his game. Over his past 40 shots, he's made 32 -much better than the \(55 \%\) he usually shoots. Do you think his chances of making a shot really increased? In other words, is making at least 32 of 40 shots really unusual for him? (Do you think it's his sneakers?)
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