28. A sample of 6500 automobiles found that 1560 of the automobiles were black, 3120 of the automobiles were sedans, and 1170 of the automobiles were black sedans. Find the probability that a randomly chosen automobile from this sample is black or a sedan.

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Step 1: Understand the problem. We are tasked with finding the probability that a randomly chosen automobile is either black or a sedan. This involves using the formula for the probability of the union of two events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B), where A represents 'black automobiles' and B represents 'sedans'.

Step 2: Calculate P(A), the probability of an automobile being black. Divide the number of black automobiles (1560) by the total number of automobiles (6500). The formula is P(A) = 1560 / 6500.

Step 3: Calculate P(B), the probability of an automobile being a sedan. Divide the number of sedans (3120) by the total number of automobiles (6500). The formula is P(B) = 3120 / 6500.

Step 4: Calculate P(A ∩ B), the probability of an automobile being both black and a sedan. Divide the number of black sedans (1170) by the total number of automobiles (6500). The formula is P(A ∩ B) = 1170 / 6500.

Step 5: Use the formula for the union of two events to find P(A ∪ B). Substitute the values of P(A), P(B), and P(A ∩ B) into the formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). This will give the probability that a randomly chosen automobile is either black or a sedan.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it helps determine the chance of selecting an automobile that is either black or a sedan from the sample. Understanding how to calculate probabilities is essential for solving the question.

Union of Events

The union of two events, denoted as A ∪ B, represents the occurrence of at least one of the events. In this case, we are interested in the probability of selecting an automobile that is either black or a sedan. The formula for the union of two events accounts for any overlap, ensuring that we do not double-count automobiles that are both black and sedans.

Counting Principles

Counting principles, such as the addition rule, are used to determine the total number of favorable outcomes when dealing with multiple categories. In this scenario, we need to count the number of black automobiles, sedans, and subtract the overlap (black sedans) to find the total number of automobiles that are either black or sedans. This is crucial for accurately calculating the probability.