Probability: Definitions and Key Concepts Explained
Probability: Definitions and Key Concepts
Probability: A numeric measure of the likelihood that an event will occur.
Impossible Event: An event that is certain not to occur. The probability of an impossible event is 0.
Certain Event: An event that is certain to occur. The probability of a certain event is 1.
Experiment: Any process that generates well-defined outcomes.
Sample Space: The set of all sample points (experimental outcomes).
Sample Point: An experimental outcome and an element of the sample space.
Basic Requirements of Probability: Two requirements that restrict the manner in which probability assignments can be made:
- For each experimental outcome E, 0 ≤ P(E) ≤ 1.
- P(E₁) + P(E₂) + · · · + P(Eₙ) = 1.
Classical Method: A method of assigning probabilities that is based on the assumption that the experimental outcomes are equally likely.
Relative Frequency Method: A method of assigning probabilities based on experimentation or historical data.
Subjective Method: A method of assigning probabilities based on judgment.
Event: A collection of sample points or experimental outcomes.
Complement of Event A: The event containing all sample points that are not in A.
Venn Diagram: A graphical device for representing the sample space and operations involving events.
Union of Events A and B: The event containing all sample points that are in A, in B, or in both.
Intersection of Events A and B: The event containing all sample points that are in both A and B.
Addition Law: A probability law used to compute the probability of a union: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0, and the addition law simplifies to P(A ∪ B) = P(A) + P(B).
Mutually Exclusive Events: Events that have no sample points in common; that is, A ∩ B is empty and P(A ∩ B) = 0.
Conditional Probability: The probability of an event given another event has occurred. The conditional probability of A given B is P(A | B) = P(A ∩ B) / P(B).
Joint Probability: The probability of the intersection of two events.
Joint Probability Table: A table used to display joint and marginal probabilities.
Marginal Probabilities: The values in the margins of the joint probability table, which provide the probability of each event separately.
Dependent Events: Two events A and B for which P(A | B) ≠ P(A) or P(B | A) ≠ P(B); that is, the probability of one event is altered or affected by knowing whether the other event occurs.
Independent Events: Two events A and B for which P(A | B) = P(A) and P(B | A) = P(B); that is, the events have no influence on each other.
Multiplication Law: A probability law used to compute the probability of an intersection: P(A ∩ B) = P(A | B)P(B) or P(A ∩ B) = P(B | A)P(A). For independent events P(A ∩ B) = P(A)P(B).
Prior Probabilities: Initial probabilities of events.
Posterior Probabilities: Revised probabilities of events based on additional information.
Bayes’ Theorem: A method used to compute posterior probabilities.
Simpson’s Paradox: The phenomenon by which the apparent association between two events is reversed upon consideration of a third event.