The addition rule helps you find the probability that event A happens, event B happens, or both happen. In statistics, the word or usually means inclusive or, so A or B includes the overlap where both events occur. This rule matters because many real situations involve combined outcomes, such as drawing a heart or a face card from a deck.
A Venn diagram makes the idea visible by showing the sample space, the two events, and their shared region.
When you add P(A) and P(B), any outcome in both events is counted once in P(A) and once again in P(B). The addition rule fixes this by subtracting P(A and B) one time. If the events are mutually exclusive, they cannot happen together, so the overlap probability is 0.
The rule is used in surveys, cards, dice, genetics, quality control, and any situation where probabilities from multiple categories must be combined carefully.
Key Facts
- General addition rule: P(A or B) = P(A) + P(B) - P(A and B)
- Intersection: P(A and B) means the probability that both A and B occur.
- Union: P(A or B) means the probability that A occurs, B occurs, or both occur.
- Mutually exclusive addition rule: If A and B cannot both occur, P(A or B) = P(A) + P(B).
- For mutually exclusive events, P(A and B) = 0.
- Probability bounds: 0 <= P(A or B) <= 1.
Vocabulary
- Sample space
- The sample space is the set of all possible outcomes of a random process.
- Event
- An event is a collection of one or more outcomes from the sample space.
- Union
- The union of A and B is the set of outcomes that are in A, in B, or in both.
- Intersection
- The intersection of A and B is the set of outcomes that are in both A and B.
- Mutually exclusive
- Mutually exclusive events are events that cannot happen at the same time.
Common Mistakes to Avoid
- Adding P(A) and P(B) without subtracting the overlap is wrong because outcomes in both events get counted twice.
- Treating or as exactly one of the events is wrong because in probability, A or B usually includes the case where both A and B occur.
- Using the mutually exclusive formula when events overlap is wrong because P(A and B) is not 0 for overlapping events.
- Reporting an answer greater than 1 is wrong because probabilities must stay between 0 and 1.
Practice Questions
- 1 In a class, P(plays soccer) = 0.40, P(plays basketball) = 0.35, and P(plays both) = 0.15. Find P(plays soccer or basketball).
- 2 From a standard 52-card deck, find the probability of drawing a heart or a king. Use that there are 13 hearts, 4 kings, and 1 king of hearts.
- 3 Events A and B are described as mutually exclusive. Explain what this tells you about their Venn diagram and how the addition rule changes.