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AP Statistics Chapter 6 Notes Probability Terms Random: Individual outcomes are uncertain, but there is a predictable distribution of outcomes in the long run. Probability: long term relative frequency Sample Space: The set of all possible outcomes of a random phenomenon. Sample space for rolling one die S = {1, 2, 3, 4, 5, 6} Sample space for the heights of adult males S = {all real x such that 30in < x < 100in} Ways to determine Sample Space 1. Tree diagram 2. Multiplication Principle: If one task can be done n1 number of ways and another can be done n2 number of ways, then both tasks can be done in n1 × n2 number of ways. 3. Organized list Events Any outcome or set of outcomes of a random phenomenon. (It is a subset of the sample space). Ex: rolling a 1 Ex: rolling a 2 or 3 Ex: Randomly choosing an adult male between 60 and 65 inches tall. Other probability terms Sampling with replacement: Each pick is the same…(number goes back in the hat). Sampling without replacement: Each draw is different. Mutually exclusive/disjoint: Two (or more) events have no outcomes in common and thus can never occur simultaneously. Complement: The complement of any event, A, is the event that A does not occur. (Ac) Basic Probability Rules 1. For any event, A, 0 < P(A) < 1. 2. If S is the sample space, then P(S) = 1. 3. Addition Rule: If A and B are disjoint, then P(A or B) = P(A U B) = P(A) + P(B) 4. Complement Rule: P(Ac) = 1 – P(A) Set Notation More Set Notation Independence Independence: Knowing that one event occurs does not change the probability that the other event occurs. 5. Multiplication Rule If events A and B are independent, then P(A and B) = P(A ∩ B) = P(A) × P(B) General Addition Rule Reminder….addition rule for mutually exclusive events is… P(A U B U C….) = P(A) + P(B) + P(C) + … The General Addition Rule applies to the union of two events, disjoint or not. P(A or B) = P(A) + P(B) – P(A and B) P(A U B) = P(A) + P(B) – P(A ∩ B) Conditional Probability P(A|B) “The probability of event A given that event B has occurred.” Examples: One card has been picked from a deck. Find… One dice has been rolled. Find… P(spade|black), P(queen|face card) P(3|odd), P(odd|prime) Two dice are rolled. Find P(2nd die is 4|1st die is 3). New definition of independence: Events A and B are independent if P(A) = P(A|B). General Multiplication Rule Reminder….Multiplication Rule for independent events is… P(A ∩ B) = P(A) × P(B) The General Multiplication Rule applies to the intersection of two events, independent or not. P(A ∩ B) = P(A) × P(B|A) P(A ∩ B) = P(B) × P(A|B) Why does this rule also work for independent events? Conditional Probability Formula Using algebra, we can rearrange the general multiplication rule to write a formula for conditional probability. P(B|A) = P(A ∩ B) ÷ P(A) P(A|B) = P(A ∩ B) ÷ P(B)