Book cover for Probability and Statistics for Engineers and Scientists

Probability and Statistics for Engineers and Scientists

Ronald E. Walpole, Raymond H. Myers, Sharon L

ISBN #9780131877115

8th Edition

1,256 Questions

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Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the motivation for using discrete probability distributions to model random variables observed in various experiments and real-life scenarios. It highlights that a few key distributions, including binomial, hypergeometric, geometric (and negative binomial), and Poisson, can effectively describe the behavior of random events, ensuring that data from diverse processes can be analyzed using standardized methods.

Learning Objectives

1

Understand the concept of discrete probability distributions and how they describe the behavior of random variables.

2

Identify and differentiate between key discrete distributions such as the binomial, hypergeometric, geometric (and negative binomial), and Poisson distributions.

3

Apply discrete probability distributions to model and analyze real-life phenomena and experimental data.

4

Interpret graphical, tabular, and formula-based representations of probability distributions.

Key Concepts

CONCEPT

DEFINITION

Discrete Probability Distribution

A function that provides the probabilities of occurrence of different outcomes for a discrete random variable.

Random Variable

A numerical outcome of a random phenomenon, which can be described using probabilities.

Binomial Distribution

A probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials with the same probability of success.

Hypergeometric Distribution

A distribution used for sampling without replacement, often applied when modeling the number of successes in a sample drawn from a finite population.

Geometric Distribution

A special case of the negative binomial distribution that models the number of trials needed to achieve the first success in repeated independent trials.

Negative Binomial Distribution

A generalization of the geometric distribution that models the number of trials needed to achieve a specified number of successes.

Poisson Distribution

A distribution that describes the probability of a given number of events occurring in a fixed interval of time or space, commonly used for modeling count data.

Example Problems

Example 1

An employee is selected from a staffof 10 to supervise a certain project by selecting a tag at random from a box containing 10 tags numbered from 1 to $10 .$ Find the formula for the probability distribution of $X$ representing the number on the tag that is drawn. What is the probability that the number drawn is less than $4 ?$

Example 2

Twelve people: are given two identical speakers to listen to for differences, if any. Suppose that these people answered by guessing only. Find the probability that three people claim to have heard a difference between the two speakers.

Example 3

Find the mean and variance of the random variable A' of Exercise 5.1 .

Example 4

In a certain city district the need for money to buy drugs is stated as the reason for $75 \%$ of all thefts. Find the probability that among the next 5 theft cases reported in this district, (a) exactly 2 resulted from the need for money to buy drugs; (b) at most 3 resulted from the need for money to buy drugs.

Example 5

According to Chemical Engineering Progress (Nov. 1990), approximately $30 \%$ of all pipework failures in chemical plants are caused by operator error. (a) What is the probability that out of the next 20 pipework failures at least 10 are due to operator error? (b) What is the probability that no more than 4 out of 20 such failures are due to operator error? (c) Suppose, for a particular plant, that out of the random sample of 20 such failures, exactly 5 are operational errors. Do you feel that the $30 \%$ figure stated above applies to this plant? Comment.

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Step-by-Step Explanations

QUESTION

How do you calculate the probability of exactly k cured patients out of n in a drug trial, assuming each patient has an independent probability p of being cured?

STEP-BY-STEP ANSWER:

Step 1: Identify that the number of cured patients follows a binomial distribution with parameters n (total patients) and p (probability of cure).
Step 2: Use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k).
Step 3: Calculate the combination (n choose k), which represents the number of ways to choose k successes from n trials.
Step 4: Substitute the values of n, k, and p into the formula.
Step 5: Compute the result to find the probability of exactly k cured patients.
Final Answer: The probability is given by P(X = k) = (nCk) * p^k * (1-p)^(n-k).

Modeling a Drug Trial with Binomial Distribution

QUESTION

How do you determine the probability of finding exactly k defective items in a sample of size n drawn from a batch containing a total of N items, with D defectives?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the scenario involves sampling without replacement, which follows a hypergeometric distribution.
Step 2: Use the hypergeometric probability formula: P(X = k) = [ (D choose k) * ((N-D) choose (n-k)) ] / (N choose n).
Step 3: Compute the combinations for selecting k defectives out of D, and n-k non-defectives out of (N-D).
Step 4: Substitute the computed values into the formula.
Step 5: Calculate the result to determine the probability of exactly k defective items in the sample.
Final Answer: The probability is given by P(X = k) = [(DCk) * ((N-D)C(n-k))] / (NCn).

Quality Control with Hypergeometric Distribution

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Common Mistakes

  • Confusing discrete probability distributions with continuous probability distributions.
  • Assuming that all random variables need unique formulas, rather than recognizing that many share similar behavior patterns.
  • Misapplying a probability distribution to a scenario that doesn't meet the required conditions, such as using a binomial model when sampling without replacement.
  • Overcomplicating the computation of probabilities by not simplifying the problem into standard models.