Set theory, Bayes theorem, independence, combinatorics and basic probability rules.
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- What is a sample space in probability?
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The set of all possible outcomes of a random experiment.- What is an event in probability?
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A subset of the sample space.- State the three axioms of probability.
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1. P(S) = 1 for sample space S; 2. P(A) >= 0 for any event A; 3. P(union of disjoint events) = sum of probabilities.- What is the union of two sets A and B?
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The set of elements that belong to A or B or both, denoted A ∪ B.- What is the intersection of two sets A and B?
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The set of elements that belong to both A and B, denoted A ∩ B.- What is the complement of a set A in sample space S?
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The set of all elements in S that are not in A, denoted A^c.- State De Morgan's first law.
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(A ∪ B)^c = A^c ∩ B^c- State De Morgan's second law.
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(A ∩ B)^c = A^c ∪ B^c- Define conditional probability P(A|B).
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P(A|B) = P(A ∩ B) / P(B), the probability of A given that B has occurred.- State Bayes theorem.
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P(A|B) = P(B|A) * P(A) / P(B)- State the law of total probability for partition ____.
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P(A) = sum of P(A|Bi) * P(Bi) for all i- When are two events A and B independent?
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When P(A ∩ B) = P(A) * P(B), or equivalently P(A|B) = P(A).- What is the multiplication rule for dependent events?
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P(A ∩ B) = P(A) * P(B|A)- What is the probability that mutually exclusive events A and B both occur?
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Zero, since P(A ∩ B) = 0.- What is a partition of a sample space?
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A collection of mutually exclusive events whose union is the entire sample space.
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