Sample Space, Events, and Basic Probability
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). The sample space (S) is the set of all possible outcomes. For a six-sided die, S = {1, 2, 3, 4, 5, 6}. An event is any subset of the sample space. The event 'rolling an even number' = {2, 4, 6}. The probability of an event A is: P(A) = (number of favorable outcomes) / (total outcomes in sample space). For rolling an even number: P(A) = 3/6 = 0.5. The complement rule states that P(not A) = 1 − P(A). The addition rule for mutually exclusive events (events that cannot both occur): P(A or B) = P(A) + P(B). For non-mutually exclusive events: P(A or B) = P(A) + P(B) − P(A and B), where P(A and B) is the probability both occur simultaneously. For drawing a card that is either a heart or a face card from a standard deck: P(heart) = 13/52, P(face card) = 12/52, P(heart and face card) = 3/52, so P(heart or face card) = 13/52 + 12/52 − 3/52 = 22/52 ≈ 0.42.