뒤로Probability: The Addition Rule and Mutually Exclusive Events
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Probability
The Addition Rule and Mutually Exclusive Events
This section explores the concept of mutually exclusive events and the Addition Rule in probability, which are fundamental for understanding how to calculate the likelihood of combined events in statistics.
Mutually Exclusive Events
Definition: Two events A and B are mutually exclusive if they cannot occur at the same time. That is, they have no outcomes in common.
Sample Space: The set of all possible outcomes for an experiment. If A and B are mutually exclusive, their intersection is empty.
Examples:
Rolling a 3 or a 4 on a die: Mutually exclusive, since a single roll cannot be both 3 and 4.
Selecting a male student or a nursing major: Not mutually exclusive, as a student can be both.
Selecting a blood donor with type O blood or a female blood donor: Not mutually exclusive, as a donor can be both.
The Addition Rule
The Addition Rule is used to find the probability that at least one of two events occurs.
General Addition Rule: For any two events A and B:
Simplified Addition Rule for Mutually Exclusive Events: If A and B are mutually exclusive:
Extension: This rule can be extended to any number of mutually exclusive events.
Examples of Using the Addition Rule
Example 1: Selecting a card from a standard deck. Probability of getting a 4 or an ace (mutually exclusive):
Example 2: Rolling a die. Probability of rolling a number less than 3 or rolling an odd number (not mutually exclusive):
Example 3: Blood bank donors. Probability donor has type O or type A blood (mutually exclusive):
Example 4: Probability donor has type B or is Rh-negative (not mutually exclusive):
Combining Rules to Find Probabilities
Complement Rule: The probability that event A does not occur is:
Application: To find the probability that a randomly selected draft pick is not a running back or wide receiver, first find the probability that they are one of these, then subtract from 1.
Summary Table: Mutually Exclusive vs. Not Mutually Exclusive Events
Scenario | Mutually Exclusive? | Reason |
|---|---|---|
Roll a 3 or 4 on a die | Yes | Cannot roll both at once |
Select a male student or a nursing major | No | Can be both |
Blood donor with type O or female donor | No | Can be both |
Card is a 4 or an ace | Yes | Cannot be both |
Key Terms
Mutually Exclusive Events: Events that cannot occur simultaneously.
Addition Rule: Formula for finding the probability of A or B occurring.
Complement: The probability that an event does not occur.
Sample Space: The set of all possible outcomes.
Applications
Calculating probabilities in games of chance (cards, dice).
Analyzing survey data (e.g., blood types, student majors).
Making predictions in business (e.g., sales volumes).
