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Chapter 2: Problem 8

Suppose that \(A\) and \(B\) are mutually exclusive events for which \(P(A)=.3\) and \(P(B)=.5 .\) What is the probability that (a) either \(A\) or \(B\) occurs? (b) \(A\) occurs but \(B\) does not? (c) both \(A\) and \(B\) occur?

Short Answer

Expert verified
(a) The probability that either \(A\) or \(B\) occurs is \(0.8\). (b) The probability that \(A\) occurs but \(B\) does not is \(0.3\). (c) The probability that both \(A\) and \(B\) occur is \(0\).

Step by step solution

01

(a) Probability of A or B occurring

To find the probability of either \(A\) or \(B\) occurring, we will use the formula for the probability of the union of two events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Since \(A\) and \(B\) are mutually exclusive, \(P(A \cap B) = 0\). Therefore, the formula simplifies to \(P(A \cup B) = P(A) + P(B)\). Using the given probabilities, we obtain: \(P(A \cup B) = P(A) + P(B) = 0.3 + 0.5 = 0.8\) So the probability that either \(A\) or \(B\) occurs is \(0.8\).
02

(b) Probability of A occurring but not B

Since A and B are mutually exclusive events, the probability of A occurring but not B is just the probability of A occurring. Therefore: \(P(A \text{ and not } B) = P(A) = 0.3\) So the probability that \(A\) occurs but \(B\) does not is \(0.3\).
03

(c) Probability of both A and B occurring

As mentioned earlier, since \(A\) and \(B\) are mutually exclusive events, they cannot both happen at the same time. Therefore, the probability that both \(A\) and \(B\) occur is 0: \(P(A \cap B) = 0\) So the probability that both \(A\) and \(B\) occur is \(0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mutually Exclusive Events
Understanding mutually exclusive events is a fundamental aspect of probability theory. These are events that cannot both happen at the same time. In other words, the occurrence of one event means the other cannot possibly occur. A simple real-world example would be flipping a coin; the events 'heads' and 'tails' are mutually exclusive because the coin cannot land on both sides at once.

When we apply this to our exercise, event A and event B are mutually exclusive with probabilities of 0.3 and 0.5, respectively. If we want to determine the probability of either A or B occurring, we merely add the individual probabilities together. This is because, by nature, their mutual exclusivity eliminates any overlap, leading to a straight sum of their probabilities.
Probability Theory
The cornerstone of probability theory is to measure how likely events are to occur. This is expressed in a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. One can think of probability as the long-term frequency of occurrence of an event when an experiment is repeated under identical conditions.

In the context of our exercise, the probability theory is used to establish the likelihood of different combinations of events A and B occurring. The solution's step-by-step approach ensures that students can follow the logical implications of the rules dictated by probability theory to reach their conclusions.
Union of Events
The union of events refers to the combined occurrence of two or more events. In probability calculation, it's symbolized by A \(\cup\) B, and it essentially asks what the chance is of either event A, event B, or both occurring. To compute this, we typically consider the individual probabilities of each event and also their intersection (the chance of both events happening together).

In the case of mutually exclusive events, the intersection is zero because the events can't happen simultaneously, simplifying the probability of their union to the sum of their probabilities. The exercise emphasizes this by showcasing that the probability of both A and B occurring in such a scenario is always zero, which leads students to properly understanding the nature of mutually exclusive events and their impact on joint probabilities.

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Most popular questions from this chapter

(a) If \(N\) people, including \(A\) and \(B,\) are randomly arranged in a line, what is the probability that \(A\) and \(B\) are next to each other? (b) What would the probability be if the people were randomly arranged in a circle?

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