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Probability Rules
Addition, Multiplication & Complement
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Probability rules help us measure how likely events are and how different events combine. They are essential in statistics, science, finance, and everyday decision making because they turn uncertainty into numbers we can analyze. A good grasp of probability rules makes it easier to interpret data, evaluate risk, and avoid common reasoning errors. Visual tools like sample spaces and Venn diagrams help students connect formulas to real event relationships.
A sample space contains all possible outcomes, and events are subsets of that space. When two events overlap, the overlap represents outcomes that belong to both events, which is written as A ∩ B. The union A ∪ B includes outcomes in A or B or both, so the overlap must be counted carefully. Probability rules such as complements, addition, and conditional probability let us calculate these regions accurately and understand whether events are independent or mutually exclusive.
Key Facts
- For any event A, 0 ≤ P(A) ≤ 1.
- The total probability of the sample space is P(S) = 1.
- Complement rule: P(A^c) = 1 - P(A).
- Addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
- If A and B are mutually exclusive, then P(A ∩ B) = 0.
- Conditional probability: P(A|B) = P(A ∩ B) / P(B), for P(B) > 0.
Vocabulary
- Sample space
- The sample space is the set of all possible outcomes of a random experiment.
- Event
- An event is any collection of outcomes from the sample space.
- Intersection
- The intersection of A and B, written A ∩ B, is the set of outcomes that are in both events.
- Union
- The union of A and B, written A ∪ B, is the set of outcomes that are in A or B or both.
- Complement
- The complement of A, written A^c, is the set of outcomes that are not in event A.
Common Mistakes to Avoid
- Adding P(A) and P(B) without subtracting the overlap, which is wrong because outcomes in A ∩ B get counted twice. Use P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
- Confusing mutually exclusive events with independent events, which is wrong because mutually exclusive means no overlap while independent means one event does not change the probability of the other. Two nonzero mutually exclusive events are not independent.
- Using conditional probability without checking the condition event, which is wrong because P(A|B) requires dividing by P(B) and only makes sense when P(B) > 0. Always identify the restricted sample space first.
- Forgetting that probabilities must stay between 0 and 1, which is wrong because any result outside this range signals an arithmetic or logic error. Recheck subtraction, overlap, and totals if this happens.
Practice Questions
- 1 In a class survey, P(A) = 0.45, P(B) = 0.35, and P(A ∩ B) = 0.15. Find P(A ∪ B) and the probability of being in neither A nor B.
- 2 A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. One marble is drawn at random. Let A be drawing a red marble and B be drawing a blue marble. Find P(A), P(B), P(A ∩ B), and P(A ∪ B).
- 3 Explain why two events with P(A ∩ B) = 0 are not necessarily independent. State what additional condition would be needed for independence.