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Chapter 5: Problem 17

In Problems 13-18, find the probability of the indicated event if \(P(E)=0.25\) and \(P(F)=0.45\) \(P\left(E^{c}\right)\)

Short Answer

Expert verified
0.75

Step by step solution

01

Understand the concept of complementary events

The probability of the complement of event E, denoted as \(E^{c}\), is the probability that event E does not occur. The sum of the probabilities of an event and its complement is always 1.
02

Use the formula for the complement

The formula for the probability of the complement of an event is given by: \[ P(E^{c}) = 1 - P(E) \]
03

Substitute the given probability

From the problem, we are given \(P(E)=0.25\). Substitute this value into the formula: \[ P(E^{c}) = 1 - 0.25 \]
04

Calculate the probability

Perform the subtraction to find the probability of the complement: \[ P(E^{c}) = 1 - 0.25 = 0.75 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complementary events
In probability theory, understanding complementary events is crucial. Complementary events refer to two outcomes of an event that cover all possibilities. For example, if event E represents 'it rains today,' the complement, denoted as \(E^{c}\), represents 'it does not rain today.'
The key point is that one of these two outcomes must happen. The sum of their probabilities is always 1. This relationship is expressed mathematically as:
\[ P(E) + P(E^{c}) = 1 \].
Therefore, the probability of the complement of event E can be calculated using:
\[ P(E^{c}) = 1 - P(E) \].
probability formula
The probability formula for complementary events helps us determine the likelihood of an event not occurring. Given the probability of event E, the formula for the complement is:
\(P(E^{c}) = 1 - P(E)\).
This formula is derived from the concept that the sum of the probabilities of all possible outcomes must equal 1. For instance, if the probability of a snowfall on a specific day, \(P(E)\), is 0.25, then the probability that it will not snow, \(P(E^{c})\), is:
\(P(E^{c}) = 1 - 0.25 = 0.75\).
Using this simple formula ensures that you can easily find the complementary probability of any event.
basic probability operations
Basic probability operations help us calculate and understand different events' probabilities. Below are key operations you should know:
  • Addition Rule: For mutually exclusive events, E and F, the probability of either event occurring is:
    \[P(E \text{ or } F) = P(E) + P(F)\]
  • Multiplication Rule: For independent events, E and F, the probability of both occurring is:
    \[P(E \text{ and } F) = P(E) \times P(F)\]
  • Complement Rule: As shown earlier, the probability that event E does not happen is:
    \[P(E^{c}) = 1 - P(E)\]

Practicing these basic operations can ease your understanding of more complex probability problems.

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Most popular questions from this chapter

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