Probability Concepts Overview

Study Notes

Rolling a die: A standard die has six sides (1-6). The sample space (all possible outcomes) is {1, 2, 3, 4, 5, 6}.
Flipping a coin: Flipping a coin twice has four possible outcomes: HH, HT, TH, TT. The sample space is {HH, HT, TH, TT}.
Probability of an event: The probability of an event (e.g., rolling a 3) is a value between 0 and 1, inclusive. A probability of 1 means the event is certain to happen, 0 means it cannot.
Mutually exclusive events (disjoint events): Events that cannot occur at the same time. The probability of A or B occurring is P(A∪B) = P(A) + P(B).
Conditional probability: The probability of event B occurring given that event A has already occurred, written as P(B|A). Calculated as P(A∩B) / P(A).
Complement of an event: The event that does not happen. P(A') = 1 - P(A), where A' is the complement of A.
Total probability rule: The probability of an event A can be found by considering all possible ways it can occur, given other events affecting or influencing it.
Bayes' theorem: Related to conditional probability, it allows calculating P(A|B) from P(B|A).

Probability of the empty set: P(Ø) = 0
Probability of the sample space: P(S) = 1
Addition rule (for mutually exclusive events): If events A and B are mutually exclusive (disjoint), then P(A or B) = P(A) + P(B). This also applies to more than two mutually exclusive events.
**Addition Rule (General case):**If A and B are not mutually exclusive, then P(A∪B) = P(A) + P(B) - P(A∩B) (the probability of both occurring).
Multiplication rule(for independent events): If events A and B are independent (the occurrence of one doesn't affect the other), then P(A and B) = P(A) * P(B)
Distributive Laws: (A∩B)∪C = (A∪C)∩(B∪C); (A∪B)∩C = (A∩C)∪(B∩C)
De Morgan's Laws: (A∪B)' = A'∩B'; (A∩B)' = A'∪B'