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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 is the set of all outcomes not in the event. The sum of the probabilities of an event and its complement is always 1.

Step by step solution

01

Define Complement of an Event

In probability theory, the complement of an event A is the set of all outcomes in the sample space that are not in A. It's often denoted as \( A' \) or \( A^c \). Therefore, if the event A is 'it will rain', its complement \( A' \) is 'it will not rain'.
02

Sum of Probabilities of Complementary Events

The sum of the probabilities of an event and its complement is always 1. This is because either the event or its complement must occur in each instance. Therefore, if \( P(A) \) is the probability of event A happening, and \( P(A') \) is the probability of its complement happening, then \( P(A) + P(A') = 1 \).

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Most popular questions from this chapter

A random sample of 250 adults was taken, and they were asked whether they prefer watching sports or opera on television. The following table gives the two-way classification of these adults. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Prefer Watching } \\ \text { Sports } \end{array} & \begin{array}{c} \text { Prefer Watching } \\ \text { Opera } \end{array} \\ \hline \text { Male } & 96 & 24 \\ \text { Female } & 45 & 85 \\ \hline \end{array} $$ a. If one adult is selected at random from this group, find the probability that this adult i. prefers watching opera ii. is a male iii. prefers watching sports given that the adult is a female iv. is a male given that he prefers watching sports \(\mathrm{v} .\) is a female and prefers watching opera vi. prefers watching sports or is a male b. Are the events "female" and "prefers watching sports" independent? Are they mutually exclusive? Explain why or why not.

Terry \& Sons makes bearings for autos. The production system involves two independent processing machines so that each bearing passes through these two processes. The probability that the first processing machine is not working properly at any time is \(.08\), and the probability that the second machine is not working properly at any time is \(.06\). Find the probability that both machines will not be working properly at any given time.

According to an Automobile Association of America report, \(9.6 \%\) of Americans traveled by car over the 2011 Memorial Day weekend and \(88.09 \%\) stayed home. What is the probability that a randomly selected American stayed home or traveled by car over the 2011 Memorial Day weekend? Explain why this probability does not equal 1.0.

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The probability that a student graduating from Suburban State University has student loans to pay off after graduation is .60. If two students are randomly selected from this university, what is the probability that neither of them has student loans to pay off after graduation?
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