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Chapter 4: Problem 101

How is the addition rule of probability for two mutually exclusive events different from the rule for two mutually nonexclusive events?

Short Answer

Expert verified
The addition rule for mutually exclusive events is the simple sum of the probabilities of each event, as these events cannot occur at the same time. For mutually nonexclusive events, which can occur concurrently, the addition rule is the sum of the probabilities of each event, subtracting the probability of their intersection, i.e., the chance of both events happening simultaneously. The difference is in accounting (or not) for the overlap between events.

Step by step solution

01

Define Mutually Exclusive Events and its Addition Rule

Mutually exclusive events are those that cannot occur at the same time. In other words, if one event happens, the other cannot. This property leads us to the addition rule for mutually exclusive events, which states that the probability of either of the events happening (event A or event B) is the sum of their individual probabilities. It can be mathematically expressed as: \( P(A \cup B) = P(A) + P(B) \).
02

Define Mutually Nonexclusive Events and its Addition Rule

On the other hand, mutually nonexclusive events are those that can happen at the same time. Under this circumstance, the addition rule involves taking into account the overlap (or intersection) between these events. The probability of either event A or event B happening is the sum of their individual probabilities minus the probability of both events A and B happening at the same time which is their intersection. This can be mathematically represented as: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).
03

Compare the Two Rules

The major difference between the addition rule of mutually exclusive and nonexclusive events lies in accounting for the intersection between the two events in the case of nonexclusive events, and lack thereof in the case of exclusive events. If events are mutually exclusive, their intersection is zero (i.e., \( P(A \cap B) = 0 \)), so the addition rule for mutually exclusive events simplifies to the addition rule for mutually nonexclusive events.

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Most popular questions from this chapter

A random sample of 80 lawyers was taken, and they were asked if they are in favor of or against capital punishment. The following table gives the two-way classification of their responses. $$\begin{array}{lcc} \hline & \begin{array}{c} \text { Favors Capital } \\ \text { Punishment } \end{array} & \begin{array}{c} \text { Opposes Capital } \\ \text { Punishment } \end{array} \\ \hline \text { Male } & 32 & 24 \\ \text { Female } & 13 & 11 \\ \hline \end{array}4\( a. If one lawyer is randomly selected from this group, find the probability that this lawyer i. favors capital punishment ii. is a female iii. opposes capital punishment given that the lawyer is a female iv. is a male given that he favors capital punishment \)\mathrm{v}\(. is a female and favors capital punishment vi. opposes capital punishment \)o r$ is a male b. Are the events "female" and "opposes capital punishment" independent? Are they mutually exclusive? Explain why or why not.

Twenty percent of a town's voters favor letting a major discount store move into their neighborhood, \(63 \%\) are against it, and \(17 \%\) are indifferent. What is the probability that a randomly selected voter from this town will either be against it or be indifferent? Explain why this probability is not equal to \(1.0\).

Five percent of all items sold by a mail-order company are returned by customers for a refund. Find the probability that, of two items sold during a given hour by this company, a. both will be returned for a refund b. neither will be returned for a refund Draw a tree diagram for this problem.

When is the following addition rule used to find the probability of the union of two events \(A\) and \(B\) ? $$P(A \text { or } B)=P(A)+P(B)$$ Give one example where you might use this formula.

What is meant by the joint probability of two or more events? Give one example.
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