AP StatisticsAP Statistics

Chapter 6 NotesChapter 6 Notes

Probability TermsProbability Terms

Random: Individual outcomes are Random: Individual outcomes are uncertain, but there is a predictable uncertain, but there is a predictable distribution of outcomes in the long run.distribution of outcomes in the long run.

Probability: long term relative frequencyProbability: long term relative frequency Sample Space: The set of all possible Sample Space: The set of all possible

outcomes of a random phenomenon.outcomes of a random phenomenon. Sample space for rolling one dieSample 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 malesSample space for the heights of adult males

S = {all real x such that 30in S = {all real x such that 30in << x x << 100in} 100in}

Ways to determine Ways to determine Sample SpaceSample Space

1. Tree diagram1. Tree diagram

2. Multiplication Principle: If one task 2. Multiplication Principle: If one task can be done ncan be done n11 number of ways and number of ways and another can be done nanother can be done n22 number of ways, number of ways, then both tasks can be done in nthen both tasks can be done in n11 × n × n22 number of ways.number of ways.

3. Organized list3. Organized list

EventsEvents

Any outcome or set of outcomes of a Any outcome or set of outcomes of a random phenomenon. (It is a subset random phenomenon. (It is a subset of the sample space).of the sample space).

Ex: rolling a 1Ex: rolling a 1 Ex: rolling a 2 or 3Ex: rolling a 2 or 3 Ex: Randomly choosing an adult Ex: Randomly choosing an adult

male between 60 and 65 inches tall.male between 60 and 65 inches tall.

Other probability termsOther probability terms Sampling Sampling withwith replacement: Each pick is replacement: Each pick is

the same…(number goes back in the hat).the same…(number goes back in the hat). Sampling Sampling withoutwithout replacement: Each replacement: Each

draw is different.draw is different. Mutually exclusive/disjointMutually exclusive/disjoint: Two (or : Two (or

more) events have no outcomes in more) events have no outcomes in common and thus can never occur common and thus can never occur simultaneously.simultaneously.

ComplementComplement: The complement of any : The complement of any event, A, is the event that A does not event, A, is the event that A does not occur. (Aoccur. (Acc))

Basic Probability RulesBasic Probability Rules 1. For any event, A, 0 1. For any event, A, 0 << P(A) P(A) << 1. 1.

2. If S is the sample space, then P(S) = 2. If S is the sample space, then P(S) = 1.1.

3. Addition Rule: If A and B are disjoint, 3. Addition Rule: If A and B are disjoint, thenthen 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(A4. Complement Rule: P(Acc) = 1 – P(A)) = 1 – P(A)

Set NotationSet Notation

More Set NotationMore Set Notation

IndependenceIndependence

Independence: Knowing that one Independence: Knowing that one event occurs does not change the event occurs does not change the probability that the other event probability that the other event occurs.occurs.

5. Multiplication Rule5. Multiplication Rule If events A and B are independent, If events A and B are independent,

thenthen P(A and B) = P(A ∩ B) = P(A) × P(B)P(A and B) = P(A ∩ B) = P(A) × P(B)

General Addition RuleGeneral Addition Rule

Reminder….addition rule for Reminder….addition rule for mutually exclusive events is…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 General Addition Rule applies to

the union of two events, disjoint or the union of two events, disjoint or not.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 ProbabilityConditional Probability

P(A|B) P(A|B) “The probability of event A given “The probability of event A given that event B has occurred.”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(2Two dice are rolled. Find P(2ndnd die is 4|1 die is 4|1stst die is die is

3).3). New definition of independence: Events A New definition of independence: Events A

and B are independent if P(A) = P(A|B).and B are independent if P(A) = P(A|B).

General Multiplication General Multiplication RuleRule

Reminder….Multiplication Rule for Reminder….Multiplication Rule for independent events is…independent events is… P(A ∩ B) = P(A) × P(B)P(A ∩ B) = P(A) × P(B)

The General Multiplication Rule The General Multiplication Rule applies to the intersection of two applies to the intersection of two events, independent or not.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 Why does this rule also work for independent events?independent events?

Conditional Probability Conditional Probability FormulaFormula

Using algebra, we can rearrange the Using algebra, we can rearrange the general multiplication rule to write a general multiplication rule to write a formula for conditional probability.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)