Discrete probability fundamentals as covered in a discrete mathematics course, distinct from a full statistics course.
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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, a collection of one or more outcomes.- Define the probability of an event A.
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The ratio of favorable outcomes to total possible outcomes, denoted P(A).- What is P(complement of A) in terms of P(A)?
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P(A^c) = 1 - P(A)- State the addition rule for the union of two events A and B.
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P(A union B) = P(A) + P(B) - P(A intersect B)- What is the probability of the union of two mutually exclusive events?
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P(A union B) = P(A) + P(B) (no intersection term, since P(A intersect B) = 0)- Define conditional probability P(A|B).
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The probability of event A occurring given that event B has occurred, equal to P(A intersect B) / P(B).- When are two events A and B independent?
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When P(A intersect B) = P(A) * P(B), or equivalently, when P(A|B) = P(A).- State the multiplication rule for any two events A and B.
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P(A intersect B) = P(A) * P(B|A) = P(B) * P(A|B)- State Bayes' theorem.
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P(A|B) = P(B|A) * P(A) / P(B)- What is the expected value of a discrete random variable X?
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E[X] = sum of (x * P(X = x)) for all values x- Define the variance of a random variable X.
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Var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2- What is a permutation of n objects taken r at a time?
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An ordered arrangement of r objects from n objects, denoted P(n,r) or nPr, equal to n! / (n-r)!- What is a combination of n objects taken r at a time?
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An unordered selection of r objects from n objects, denoted C(n,r) or nCr, equal to n! / (r! * (n-r)!)- State the binomial probability formula.
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P(X = k) = C(n,k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success on each trial.
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