In Exercises 19-22, determine whether the events are independent ...

4. Probability

Multiplication Rule: Independent Events

Statistics

4. Probability

Multiplication Rule: Independent Events

1:31 minutes

Problem 3.RE.22

Textbook Question

In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.
22. Getting high grades and being awarded an academic scholarship

Verified step by step guidance

Understand the definition of independent and dependent events: Two events are independent if the occurrence of one event does not affect the probability of the other event occurring. Conversely, two events are dependent if the occurrence of one event affects the probability of the other event occurring.

Identify the two events in the problem: Event A is 'getting high grades,' and Event B is 'being awarded an academic scholarship.'

Analyze the relationship between the two events: Consider whether achieving high grades (Event A) influences the likelihood of being awarded an academic scholarship (Event B).

Recognize that academic scholarships are often awarded based on academic performance, which means that getting high grades (Event A) increases the probability of being awarded a scholarship (Event B).

Conclude that the events are dependent because the occurrence of Event A (getting high grades) directly affects the likelihood of Event B (being awarded an academic scholarship).

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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, flipping a coin and rolling a die 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 independent or dependent, as it helps assess how the occurrence of one event influences the likelihood of another.

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In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.22. Getting high grades and being awarded an academic scholarship