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Chapter 7: Problem 52

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(E\) is an event of an experiment, then \(P(E)+P\left(E^{c}\right)=1\).

Short Answer

Expert verified
The statement is true. The rule of complementary events states that the probabilities of an event and its complement must sum up to \(1\). Mathematically, this can be written as \(P(E) + P(E^c) = 1\), where \(E\) is an event of an experiment and \(E^c\) is its complement. Since \(E\) and \(E^c\) cover all the possible outcomes in the sample space, their probabilities must sum up to \(1\).

Step by step solution

01

Understanding Event and its Complement

Let \(E\) be an event of an experiment. Then, we can define its complement, denoted by \(E^c\), as the set of all outcomes in the sample space that are not in \(E\). In other words, when the experiment takes place, either \(E\) occurs or \(E^c\) occurs, but not both simultaneously.
02

Rule of Complementary Events

The probabilities of complementary events sum up to \(1\). In mathematical terms, this can be written as: \[P(E) + P(E^c) = 1\]
03

Explaining the Statement

The given statement states that if \(E\) is an event of an experiment, then \(P(E) + P(E^c) = 1\). This statement is true based on the rule of complementary events, which we stated earlier. Since \(E\) and \(E^c\) cover all the possible outcomes in the sample space (meaning one of them will always occur), their probabilities must sum up to \(1\).

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