Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 3 - Probability

Problem 3.2.1a

Chapter 3, Problem 3.2.1a

"1. What is the difference between independent and dependent events?

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Independent events are events where the occurrence of one event does not affect the probability of the other event occurring. For example, flipping a coin and rolling a die are independent events because the outcome of the coin flip does not influence the outcome of the die roll.

Dependent events are events where the occurrence of one event affects the probability of the other event occurring. For example, drawing two cards from a deck without replacement is a dependent event because the outcome of the first draw changes the probabilities for the second draw.

To mathematically express independence, two events A and B are independent if and only if P(A ∩ B) = P(A) × P(B), where P(A ∩ B) is the probability of both events occurring, and P(A) and P(B) are the probabilities of each event individually.

For dependent events, the probability of one event occurring given that another event has occurred is expressed using conditional probability: P(A | B) = P(A ∩ B) / P(B), where P(A | B) is the probability of A given B, P(A ∩ B) is the joint probability, and P(B) is the probability of B.

Understanding the difference between independent and dependent events is crucial for solving problems involving probability, as it determines whether you can multiply probabilities directly or need to account for conditional relationships.

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Key Concepts

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

Independent Events

Independent events are those whose outcomes do not affect each other. In probability, two events A and B are independent if the occurrence of A does not change the likelihood of B occurring, and vice versa. For example, flipping a coin and rolling a die are independent events; the result of one does not influence the other.

Dependent Events

Dependent events are events where the outcome of one event affects the outcome of another. In probability, two events A and B are dependent if the occurrence of A changes the likelihood of B occurring. For instance, drawing cards from a deck without replacement creates dependent events, as the first draw alters the composition of the deck for the second draw.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. It quantifies uncertainty and is foundational in statistics. For independent events, the probability of both events occurring is the product of their individual probabilities, while for dependent events, the probability must account for the influence of the first event on the second.

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Related Practice

Textbook Question

50. Investment Committee A company has 200 employees, consisting of 144 women and 56 men. The company wants to select five employees to serve as an investment committee.

a. Use technology to find the number of ways that 5 employees can be selected from 200.

Textbook Question

5. Which event(s) in Exercise 4 can be considered unusual? Explain your reasoning.

Textbook Question

Finding Conditional Probabilities In Exercises 7 and 8, use the table to find each conditional probability.

7. Business Degrees The table shows the numbers of male and female students in the United States who received bachelor's degrees in business and nonbusiness fields in a recent year. (Source: National Center for Educational Statistics)

a. Find the probability that a randomly selected bachelor's degree-earning student is male, given that the degree is in business.

Textbook Question

66. Access Code An access code consists of six characters. For each character, any letter or number can be used, with the exceptions that the first character cannot be 0 and the last two characters must be odd numbers.

a. What is the probability of randomly selecting the correct access code on the first try?

Textbook Question

"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.

26. Worst President In a sample of 1500 adult U.S. citizens, 690 said that Donald Trump was the worst president in U.S. history. Three adult U.S. citizens are selected at random.

(Adapted from YouGov)

a. Find the probability that all three adult U.S. citizens say that Donald Trump was the worst president in U.S. history."

Textbook Question

2. How many possible variations are there in Mozart's Musical Dice Game minuet? Explain.