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Ch. 3 - Probability
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 3 - ProbabilityProblem 3.2.13
Chapter 3, Problem 3.2.13

"Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
13. Rolling a six-sided die and then rolling the die a second time so that the sum of the two rolls is five"

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1
Understand the concept of independent and dependent events: Independent events are those where the outcome of one event does not affect the outcome of the other. Dependent events are those where the outcome of one event influences the outcome of the other.
Analyze the problem: Rolling a six-sided die twice involves two separate events. The first roll and the second roll are distinct actions.
Consider the condition provided: The sum of the two rolls must equal five. This introduces a dependency because the outcome of the first roll directly affects the possible outcomes of the second roll to satisfy the condition.
Explain the reasoning: Since the sum of the two rolls is constrained to equal five, the second roll depends on the result of the first roll. For example, if the first roll is 2, the second roll must be 3 to meet the condition.
Conclude: The events are dependent because the condition (sum equals five) creates a relationship between the outcomes of the two rolls.

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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 probability of B occurring, and vice versa. For example, rolling a die and flipping a coin are independent events because the result of one does not influence the other.
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Probability of Multiple Independent Events

Dependent Events

Dependent events are those 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 probability of B occurring. For instance, drawing cards from a deck without replacement is a classic example of dependent events, as the first draw affects the composition of the deck for the second draw.
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Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which represents the probability of event A occurring given that event B has occurred. Understanding conditional probability is crucial for determining whether events are dependent, as it helps to assess how the occurrence of one event influences the likelihood of another.
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Related Practice
Textbook Question

Using a Pie Chart to Find Probabilities In Exercises 83-86, use the pie chart at the left, which shows the number of workers (in millions) by occupation for the United States. (Source: U.S. Bureau of Labor Statistics)

84. Find the probability that a worker chosen at random is not employed in a service occupation.

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Textbook Question

34. Lottery Number Selection A lottery has 52 numbers. In how many different ways can six of the numbers be selected? (Assume the order of selection is not important.)

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Textbook Question

Boy or Girl? In Exercises 71-74, a couple plans to have three children. Each child is equally likely to be a boy or a girl.

71. What is the probability that all three children are girls?

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Textbook Question

"Classifying Events Based on Studies In Exercises 15-18, identify the two events described in the study. Do the results indicate that the events are independent or dependent? Explain your reasoning.

16. A study found no significant association between the use of talc powder and the incidence of ovarian cancer in women. (Source: JAMA)"

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Textbook Question

According to Bayes’ Theorem, the probability of event A , given that event B has occurred, is

P(A|B) = P(A) * P(B|A)P(A) * P(B|A) + P(A') * P(B|A').

In Exercises 33–38, use Bayes’ Theorem to find P(A|B).

37. P(A) = 73%, P(A') = 17%, P(B|A) = 46% , and P(B|A') = 52%

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Textbook Question

25. Playlist A band is preparing a setlist of 21 songs for a concert. How many different ways can the band play the first six songs?

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