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Chapter 10: Problem 59

Explain how to find the probability of an event not occurring. Give an example.

Short Answer

Expert verified
To find the probability of an event not occurring, subtract the probability of the event from 1. For example, if the probability of rolling a 3 on a die is \(1/6\), then the probability of not rolling a 3 is 1 - \(1/6\) = \(5/6\).

Step by step solution

01

Understanding Probability

Probability is a mathematical way of expressing the likelihood that a specific event will occur. It is expressed as a number between 0 and 1. An event with a probability of 1 means it is guaranteed to occur, while an event with a probability of 0 means it will not occur. Thus, any event has a probability \(P(A)\) between 0 and 1.
02

Understanding Complementary Events

In probability theory, every event \(A\) has a complementary event, often denoted as \(A'\) or not \(A\). The complement of an event includes all outcomes in the sample space that are not included in the original event. We need to know that \(P(A)\) + \(P(A')\) = 1.
03

Finding the Probability of an Event Not Occurring

As we have already learned that \(P(A)\) + \(P(A')\) = 1. Therefore, we can find out the probability of an event not occurring, which is the complement of the event, by subtracting the probability of the event from 1. So, the probability of an event not occurring or \(P(A')\) = 1 - \(P(A)\).
04

Giving an example

For instance, suppose we roll a fair, six-sided die. The probability of rolling a 3 (\(P(A)\)) is \(1/6\). To find the probability of not rolling a 3 (\(P(A')\)), subtract the probability of rolling a 3 from 1. So, \(P(A')\) = 1 - \(1/6\) = \(5/6\). Hence, the probability of not rolling a 3 is \(5/6\).

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