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Chapter 9: Problem 26

What is the sum of the probabilities of an event and its complement?

Short Answer

Expert verified
The sum is 1.

Step by step solution

01

Understanding probability

Probability of an event is a measure of the likelihood that the event will occur. It is a value between 0 and 1.
02

Defining complement

The complement of an event includes all outcomes that are not part of the event. If the event is A, its complement is denoted as A'.
03

Sum of probabilities rule

According to the rules of probability, the sum of the probabilities of an event A and its complement A' is always 1. Mathematically, it can be written as: \[ P(A) + P(A') = 1 \].

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Key Concepts

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

Event Probability
Event probability measures how likely an event is to happen. It is always expressed as a number between 0 and 1.
For example, if you roll a die, the probability of rolling a number less than 4 is 3 out of 6 possible outcomes, or 0.5.
In mathematical terms, the probability of event A occurring is noted as \( P(A) \).

Key points to remember:
  • Probability values range from 0 (impossible event) to 1 (certain event).
  • A probability near 0 means the event is unlikely to occur.
  • A probability near 1 means the event is likely to occur.
Complement of an Event
The complement of an event includes all the outcomes that are not part of the event. If you have event A, the complement is denoted as \( A' \).
Imagine a scenario where you flip a coin. If event A is getting heads, then the complement \( A' \) is getting tails.
The complement is everything that makes event A not happen.

Why is this important?
  • Understanding the complement helps in calculating the likelihood of an event not happening.
  • It also ties into the rule that the sum of the probabilities of an event and its complement equals 1.
Sum of Probabilities Rule
This rule states that the sum of the probabilities of an event A and its complement \( A' \) is always 1.
Mathematically: \[ P(A) + P(A') = 1 \]
This makes sense because either an event happens or it doesn't. Together, these cover all possible outcomes.

Example:
  • If the probability of raining tomorrow (event A) is 0.3, the probability of it not raining (complement \( A' \)) must be 0.7 because 0.3 + 0.7 = 1.
Remember, the key takeaway here is that the sum of the probabilities of all possible outcomes in a situation always equals 1.

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