Complement Rule

Definition

The complement rule is a fundamental concept in probability theory that states the probability of an event occurring is equal to 1 minus the probability of the event not occurring. It establishes a relationship between the probability of an event and its complement, which is the event that the original event does not occur.

5 Must Know Facts For Your Next Test

1. The complement rule states that the probability of an event occurring, P(A), plus the probability of the event not occurring, P(not A), must equal 1.
2. The complement rule is particularly useful in the context of independent and mutually exclusive events, as it allows for the calculation of probabilities without the need for detailed information about the sample space.
3. When events are mutually exclusive, the probability of one event occurring is the complement of the probability of the other event occurring.
4. The complement rule can be expressed mathematically as: P(not A) = 1 - P(A), where P(A) is the probability of the event occurring.
5. Understanding the complement rule is crucial for solving probability problems, as it provides a way to determine the probability of an event not occurring, which is often necessary for calculating the probability of other events.

Review Questions

• Explain how the complement rule is used to calculate the probability of an event not occurring.
  • The complement rule states that the probability of an event not occurring, P(not A), is equal to 1 minus the probability of the event occurring, P(A). This relationship can be expressed mathematically as P(not A) = 1 - P(A). By using the complement rule, you can determine the probability of an event not occurring without needing to know the specific details of the sample space. This is particularly useful when working with independent and mutually exclusive events, as the complement rule allows you to calculate probabilities without having to consider the entire set of possible outcomes.
• Describe the relationship between the complement rule and mutually exclusive events.
  • When events are mutually exclusive, the occurrence of one event prevents the occurrence of the other event. In this case, the probability of one event occurring is the complement of the probability of the other event occurring. For example, if the probability of event A occurring is P(A), then the probability of event A not occurring, which is the complement of event A, is P(not A) = 1 - P(A). This relationship is a direct application of the complement rule and is particularly useful when working with mutually exclusive events, as it allows you to calculate probabilities without needing to know the details of the entire sample space.
• Analyze how the complement rule can be used to solve probability problems involving independent events.
  • The complement rule is a powerful tool for solving probability problems, especially when dealing with independent events. Since independent events do not affect each other's probabilities, the complement rule can be used to calculate the probability of one event not occurring, which is often necessary for determining the probability of other events. By applying the complement rule, P(not A) = 1 - P(A), you can find the probability of an event not occurring and then use that information to calculate the probabilities of other independent events. This approach simplifies the problem-solving process and allows you to determine the overall probability of a sequence of independent events without needing to know the details of the entire sample space.

Related terms

Probability: The likelihood or chance of an event occurring, typically expressed as a number between 0 and 1.

Event: A possible outcome or occurrence in an experiment or situation.

Mutually Exclusive Events: Events that cannot occur simultaneously, meaning the occurrence of one event prevents the occurrence of the other.