Probability questions often ask about two events happening together or about at least one of two events happening. The multiplication and addition rules help answer these questions, while independence and mutual exclusivity determine which simplified versions of the rules apply.
P(A∣B)=P(B)P(A AND B)
The conditional probability P(A∣B) can differ from the overall probability P(A). If knowing that B occurred changes the probability of A, the events are dependent.
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What is the multiplication rule?
P(A AND B)=P(A∣B)P(B). It calculates the probability that both events occur.
What does P(A∣B) mean?
The probability that event A occurs given that event B has occurred.
How does independence simplify the multiplication rule?
For independent events, P(A∣B)=P(A), so P(A AND B)=P(A)P(B).
What is the addition rule?
P(A OR B)=P(A)+P(B)−P(A AND B). Subtract the overlap to avoid counting it twice.
What are mutually exclusive events?
Events that cannot occur at the same time; their intersection probability is zero.
Quiz questions
What probability does the multiplication rule calculate?Answer: The probability that both events occur.
When are two events independent?Answer: When one event's occurrence does not change the probability of the other.
Why does the addition rule subtract P(A AND B)?Answer: The overlap is included in both individual event probabilities.
A customer orders pizza with probability 0.87, and the probability of ordering salad given a pizza order is 0.55. What is the probability the customer orders both?Answer: 0.55(0.87)=0.4785.
A person can choose at most one of two vacations. If the probabilities of choosing them are 0.60 and 0.35, what is the probability of choosing either vacation?Answer: 0.95, the sum of the two probabilities.
Based on Introductory Statistics 2e by Barbara Illowsky, Susan Dean (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.