"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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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.

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.

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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