AP Statistics Chapter 6 Notes

Probability Terms Random: Individual outcomes are uncertain, but there is a predictable distribution of outcomes in the long run. Random: Individual outcomes are uncertain, but there is a predictable distribution of outcomes in the long run. Probability: long term relative frequency Probability: long term relative frequency Sample Space: The set of all possible outcomes of a random phenomenon. Sample Space: The set of all possible outcomes of a random phenomenon. Sample space for rolling one die Sample space for rolling one die S = {1, 2, 3, 4, 5, 6} S = {1, 2, 3, 4, 5, 6} Sample space for the heights of adult males Sample space for the heights of adult males S = {all real x such that 30in < x < 100in} S = {all real x such that 30in < x < 100in}

Ways to determine Sample Space 1. Tree diagram 1. Tree diagram 2. Multiplication Principle: If one task can be done n 1 number of ways and another can be done n 2 number of ways, then both tasks can be done in n 1 × n 2 number of ways. 2. Multiplication Principle: If one task can be done n 1 number of ways and another can be done n 2 number of ways, then both tasks can be done in n 1 × n 2 number of ways. 3. Organized list 3. Organized list

Events Any outcome or set of outcomes of a random phenomenon. (It is a subset of the sample space). 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 1 Ex: rolling a 2 or 3 Ex: rolling a 2 or 3 Ex: Randomly choosing an adult male between 60 and 65 inches tall. 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 with replacement: Each pick is the same…(number goes back in the hat). Sampling without replacement: Each draw is different. 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. 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. (A c ) Complement: The complement of any event, A, is the event that A does not occur. (A c )

Basic Probability Rules 1. For any event, A, 0 < P(A) < For any event, A, 0 < P(A) < If S is the sample space, then P(S) = If S is the sample space, then P(S) = Addition Rule: If A and B are disjoint, then 3. Addition Rule: If A and B are disjoint, then P(A or B) = P(A U B) = P(A) + P(B) P(A or B) = P(A U B) = P(A) + P(B) 4. Complement Rule: P(A c ) = 1 – P(A) 4. Complement Rule: P(A c ) = 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. Independence: Knowing that one event occurs does not change the probability that the other event occurs. 5. Multiplication Rule 5. Multiplication Rule If events A and B are independent, then If events A and B are independent, then P(A and B) = P(A ∩ B) = P(A) × P(B) P(A and B) = P(A ∩ B) = P(A) × P(B)

General Addition Rule Reminder….addition rule for mutually exclusive events is… Reminder….addition rule for mutually exclusive events is… P(A U B U C….) = P(A) + P(B) + P(C) + … 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. 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 or B) = P(A) + P(B) – P(A and B) P(A U B) = P(A) + P(B) – P(A ∩ 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.” P(A|B)  “The probability of event A given that event B has occurred.” Examples: Examples: One card has been picked from a deck. Find… One card has been picked from a deck. Find… P(spade|black), P(queen|face card) P(spade|black), P(queen|face card) One dice has been rolled. Find… One dice has been rolled. Find… P(3|odd), P(odd|prime) P(3|odd), P(odd|prime) Two dice are rolled. Find P(2 nd die is 4|1 st die is 3). Two dice are rolled. Find P(2 nd die is 4|1 st die is 3). New definition of independence: Events A and B are independent if P(A) = P(A|B). 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… Reminder….Multiplication Rule for independent events is… P(A ∩ B) = P(A) × P(B) P(A ∩ B) = P(A) × P(B) The General Multiplication Rule applies to the intersection of two events, independent or not. 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(A) × P(B|A) P(A ∩ B) = P(B) × P(A|B) P(A ∩ B) = P(B) × P(A|B) Why does this rule also work for independent events? 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. 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(B|A) = P(A ∩ B) ÷ P(A) P(A|B) = P(A ∩ B) ÷ P(B) P(A|B) = P(A ∩ B) ÷ P(B)