(57) AP Statistics: Probability RULES!!!!!!!!!!!!!!!!!!!!!!!!

Rule 1: The probability of an event is always between 0 and 1.

Rule 2: The sum of the probabilities of all possible outcomes of a random experiment equals 1.

Rule 3: For mutually exclusive events, the probability of either event occurring is the sum of their individual probabilities.

Rule 4: For independent events, the probability of both events occurring is the product of their individual probabilities. Rule 5: The probability of the complement of an event occurring is equal to 1 minus the probability of the event itself. Rule 6: If two events are not mutually exclusive, the probability of either event occurring is the sum of their individual probabilities minus the probability of both events occurring. Rule 7: The Law of Total Probability states that the total probability of an event can be found by considering all possible ways that event can occur. This involves summing the probabilities of each mutually exclusive scenario that leads to the event. Rule 8: Conditional probability describes the likelihood of an event occurring given that another event has already occurred, and is calculated using the formula P(A|B) = P(A and B) / P(B), where P(A|B) is the probability of event A occurring given event B. Rule 9: The multiplication rule for independent events states that the probability of both events occurring is the product of their individual probabilities, expressed as P(A and B) = P(A) * P(B) when A and B are independent. Rule 10: The complement rule states that the probability of an event not occurring is equal to one minus the probability of the event occurring, represented mathematically as P(A') = 1 - P(A). Rule 11: The addition rule for mutually exclusive events states that the probability of either event A or event B occurring is the sum of their individual probabilities, expressed as P(A or B) = P(A) + P(B) when A and B cannot happen at the same time. Rule 12: The total probability rule allows us to find the probability of an event based on a partition of the sample space, expressed as P(A) = \sum P(A|B_i) * P(B_i) for all mutually exclusive events B_i that partition the sample space. Rule 13: The Bayes' theorem provides a way to update the probability of an event based on new evidence, formulated as P(A|B) = [P(B|A) * P(A)] / P(B), allowing us to revise our predictions as more information becomes available. Rule 14: The law of total probability states that if you have a set of mutually exclusive events that collectively cover the entire sample space, then the probability of any event can be calculated by summing the probabilities of that event occurring given each of the mutually exclusive events, expressed as P(A) = \sum P(A|B) * P(B), which is useful for breaking down complex probability scenarios. Rule 15: The multiplication rule for independent events states that the probability of both event A and event B occurring is the product of their individual probabilities, represented as P(A and B) = P(A) * P(B) when A and B do not influence each other. Rule 16: The complement rule states that the probability of the complement of an event A, which is the event that A does not occur, is given by P(A') = 1 - P(A), highlighting the relationship between an event and its complement. Rule 17: The addition rule for mutually exclusive events states that the probability of either event A or event B occurring is the sum of their individual probabilities, represented as P(A or B) = P(A) + P(B), applicable when A and B cannot occur simultaneously. Rule 18: The addition rule for non-mutually exclusive events extends the previous rule by accounting for the overlap between events A and B, expressed as P(A or B) = P(A) + P(B) - P(A and B), ensuring we do not double-count the probability of both events occurring together. Rule 19: The conditional probability rule specifies that the probability of event A occurring given that event B has occurred is represented as P(A|B) = P(A and B) / P(B), which is essential for understanding how the occurrence of one event affects the likelihood of another.