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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 3.2.9
Chapter 3, Problem 3.2.9

"Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
9. Selecting a king from a standard deck of 52 playing cards, replacing it, and then selecting a queen from the deck"

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Understand the definition 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: The first event is selecting a king from a standard deck of 52 cards. The second event is selecting a queen from the same deck, but the key detail is that the king is replaced back into the deck before the second draw.
Consider the replacement: Since the king is replaced back into the deck, the total number of cards in the deck remains 52 for the second draw. This means the probability of selecting a queen is not affected by the outcome of the first draw.
Conclude independence: Because the replacement ensures that the outcome of the first event (selecting a king) does not influence the outcome of the second event (selecting a queen), the two events are independent.
Summarize reasoning: The events are independent because the replacement of the card ensures that the probabilities for the second event remain unchanged regardless of the outcome of the first event.

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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. For example, flipping a coin and rolling a die 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, 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 creates dependent events, as the first draw alters the composition of the deck for the second draw.
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Multiplication Rule: Dependent Events

Replacement in Probability

Replacement in probability refers to the practice of returning an item to its original state before the next event occurs. In the context of card selection, if a card is drawn and then replaced, the total number of cards remains constant, making subsequent selections independent. This contrasts with drawing without replacement, where the total changes and events become dependent.
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Introduction to Probability
Related Practice
Textbook Question

Finding the Probability of the Complement of an Event The age distribution of the residents of Ithaca, New York, is shown at the left. In Exercises 59-62, find the probability of the event. (Source: U.S. Census Bureau)

61. Event C: A randomly chosen resident of Ithaca is not less than 18 years old.

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Textbook Question

Finding the Probability of the Complement of an Event In Exercises 17-20, the probability that an event will happen is given. Find the probability that the event will not happen.

19. P(E)=0.03

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Textbook Question

Finding the Probability of an Event In Exercises 21-24, the probability that an event will not happen is given. Find the probability that the event will happen. 

21. P(E') =0.95

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Textbook Question

Writing In Exercises 89 and 90, write a statement that represents the complement of the probability.

90. The probability of randomly choosing a car with more than one cause for showing its "CHECK ENGINE" light from the population of vehicles showing "CHECK ENGINE" lights.

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Textbook Question

Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.

14. A ball is selected from a bin of balls numbered from 1 through 52. It is replaced, and then a second numbered ball is selected from the bin.

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Textbook Question

Board of Directors The University of Colorado Board of Directors has 23 members. One member serves as board chair and another serves as vice chair. Given the names of the 23

board members, what is the probability of randomly selecting the name of the chair and the name of the vice chair? (Source: University of Colorado)

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