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This comprehensive recap covers essential vocabulary and concepts in probability, including the likelihood of an event, sample space, events, complements, unions, intersections, independent and dependent events, and conditional probability. Learn the distinction between independent and mutually exclusive events, and how to calculate probabilities using these principles. Whether you are studying for exams or just want to strengthen your grasp of probability, this summary provides a clear and concise overview of key terms and their applications in various scenarios.
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Vocabulary • Probability • The likelihood of an event • Theoretical is what should happen. • Experimental is what does happen. • Sample Space • The set of all possible outcomes • Event • A subset of the sample space
Vocabulary • Complement of an Event • When an event does NOT occur • If A is the event that it rains, A’ is the event that it does not rain • Can be noted A’, Ac, A, or ~A
The Complement of a Set The complement of a set A is defined as the set of elements that are contained in U, the universal set, but not contained in set A. The symbolism and notation for the complement of set A are In the Venn diagram on the left, the rectangle represents the universe. A is the shaded area outside the set A.
Union of Sets The union of two sets A and B is the set of all elements formed by combining all the elements of set A and all the elements of set B into one set. It is written AB. A OR B In the Venn diagram on the left, the union of A and B is the entire region shaded. B A
Intersection of Sets The intersection of two sets A and B is the set of all elements that are common to both A and B. It is written A B. A AND B In the Venn diagram on the left, the intersection of A and B is the shaded region. B A
Disjoint Sets If two sets have no elements in common, they are said to be disjoint.Two sets A and B are disjoint if A B = . Example: The rational and irrational numbers are disjoint.
Vocabulary • Independent Events • Events are independent if the probability of one does not affect the probability of the other • Dependent Events • Events are dependent if the probability of one DOES affect the probability of the other • Mutually Exclusive (a.k.a. Disjoint) • Events are mutually exclusive if they cannot occur at the same time
What’s the Difference Between Independent and Mutually Exclusive? Mutually Exclusive:Two Events, One Trial. Rolling a 2 and rolling a 4 on one roll of a die. Independent: Two events, Two Trials. Rolling a 2 and then rolling a 4. Sometimes you need to use the conditional probability formula to tell whether events are independent or not.
Conditional Probability • Always for two events or trials P(A and B) = P(A) * P(B|A) If the trials are independent, then P(B|A) = P(B)
Relate: P( male ) = 58% P( male and jogs ) = 20% P( A and B ) P( A ) Write: P(B|A) = 0.2 0.58 = Substitute 0.2 for P(A and B) and 0.58 for P(A). 0.344 Simplify. Conditional Probability Researchers asked people who exercise regularly whether they jog or walk. Fifty-eight percent of the respondents were male. Twenty percent of all respondents were males who said they jog. Find the probability that a male respondent jogs. Define: Let A = male. Let B = jogs. The probability that a male respondent jogs is about 34%.
