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Chapter 4: Problem 44

What is the complement of an event? What is the sum of the probabilities of two complementary events?

Short Answer

Expert verified
The complement of an event includes all outcomes not in the given event. The sum of the probabilities of an event and its complement is always equal to 1.

Step by step solution

01

Understanding the Complement of an Event

In the field of probability, a complementary event of a particular event is an event that constitutes all possible outcomes that are not included in the given event. If one represents event A, then its complement is typically denoted as A'.
02

Probability of an Event and Its Complement

The total probability for all possible outcomes is 1. Therefore, the sum of the probabilities for an event A and its complement A' would sum up to 1. Since the event A and event A' are the only possibilities, it's given mathematically as P(A) + P(A') = 1.
03

Derivation of the Probability of the Complement of an Event

Knowing that the sum of the probabilities for an event A and its complement A' sums up to 1, the probability of the complement of an event can be derived by subtracting the probability of the event from 1. It's written as P(A') = 1 - P(A). This indicates that the more likely an event, the less likely it is that the event will not occur, and vice versa.

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Most popular questions from this chapter

What is the joint probability of two mutually exclusive events? Give one example.

Five hundred employees were selected from a city's large private companies, and they were asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared $$\begin{array}{llc} \hline & \text { Yes } & \text { No } \\ \hline \text { Men } & 225 & 75 \\ \text { Women } & 150 & 50 \\ \hline \end{array}$$ Suppose one employee is selected at random from these 500 employees. Find the following probabilities. a. The probability of the union of events "woman" and "yes" b. The probability of the union of events "no" and "man'

Find the joint probability of \(A\) and \(B\) for the following. a. \(P(A)=.40\) and \(P(B \mid A)=.25\) b. \(P(B)=.65\) and \(P(A \mid B)=.36\)

Jason and Lisa are planning an outdoor reception following their wedding. They estimate that the probability of bad weather is .25, that of a disruptive incident (a fight breaks out, the limousine is late, etc.) is 15 , and that bad weather and a disruptive incident will occur is .08. Assuming these estimates are correct, find the probability that their reception will suffer bad weather or a disruptive incident.

Given that \(P(A \mid B)=.40\) and \(P(A\) and \(B)=.36\), find \(P(B)\).
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