YMS Chapter 6YMS Chapter 6

Probability: Foundations for Probability: Foundations for InferenceInference

6.1 – The Idea of Probability6.1 – The Idea of Probability

Probability – 3 InterpretationsProbability – 3 Interpretations►Any outcome of any random Any outcome of any random

phenomenon is the proportion of times phenomenon is the proportion of times it would occur in a very long series of it would occur in a very long series of repetitionsrepetitions

►Long-term relative frequency Long-term relative frequency ►Branch of math that describes the Branch of math that describes the

pattern of chance outcomes pattern of chance outcomes

► When individual outcomes are uncertain When individual outcomes are uncertain but there is still a regular distribution of but there is still a regular distribution of outcomes in the long runoutcomes in the long run

► Relative frequencies of outcomes seem to Relative frequencies of outcomes seem to settle down to fixed values in the long runsettle down to fixed values in the long run

► Chance behavior is unpredictable in the Chance behavior is unpredictable in the short run but has a regular and predictable short run but has a regular and predictable pattern in the long runpattern in the long run

Randomness – 3 InterpretationsRandomness – 3 Interpretations

Exploring RandomnessExploring Randomness

► Must have a long series of independent Must have a long series of independent trialstrials

► Probability is empirical (based on previous Probability is empirical (based on previous experience)experience)

► Computer simulations are very usefulComputer simulations are very useful

6.1 Practice – p334 #6.4, 6.9 and 6.106.1 Practice – p334 #6.4, 6.9 and 6.10

YMS 6.2YMS 6.2

Probability ModelsProbability Models

► Sample space (S) Sample space (S) The set of all possible outcomesThe set of all possible outcomes

► EventEvent Any outcome or set of outcomes of a random Any outcome or set of outcomes of a random

phenomenonphenomenon A subset of SA subset of S

► Probability modelProbability model A mathematical description of a random A mathematical description of a random

phenomenon consisting of two parts: S and the phenomenon consisting of two parts: S and the assignment of probabilities to eventsassignment of probabilities to events

► Multiplication (Counting) PrincipleMultiplication (Counting) Principle Multiply number of outcomes for each event to Multiply number of outcomes for each event to

find total number of waysfind total number of ways Use a tree diagram to visually represent and find Use a tree diagram to visually represent and find

sample spacesample space

► Any probability is a number between 0 and Any probability is a number between 0 and 1.1. 0 0 << P(A) P(A) << 1 1

► All possible outcomes together must have All possible outcomes together must have probability 1.probability 1. P(S)P(S) = 1 = 1

► Complement Rule - The probability that an Complement Rule - The probability that an event does not occur is 1 minus the event does not occur is 1 minus the probability that it does. probability that it does. P(AP(ACC)) = 1- = 1- P(A)P(A)

Probability RulesProbability Rules

►Disjoint or Mutually Exclusive EventsDisjoint or Mutually Exclusive Events If two events have no outcomes in common, If two events have no outcomes in common,

the probability that one or the other occurs the probability that one or the other occurs is the sum of their individual probabilities. is the sum of their individual probabilities.

P(A or B)P(A or B) = = P(A) + P(B) – P(A and B)P(A) + P(B) – P(A and B)

► Independent eventsIndependent events Knowing that if one event occurs it does not Knowing that if one event occurs it does not

change the probability that the other occurschange the probability that the other occurs P(A and B)P(A and B) = = P(A)P(B)P(A)P(B)

Basic Set TheoryBasic Set Theory► UnionUnion

Combination of elements in setsCombination of elements in sets► IntersectionIntersection

What the sets have in common What the sets have in common ► Null setNull set

Set without elementsSet without elements► Venn Diagrams Venn Diagrams

Very useful to create when answering questions Very useful to create when answering questions about relationships among setsabout relationships among sets

6.2 Practice – p340 #6.15, 6.19, 6.26,6.2 Practice – p340 #6.15, 6.19, 6.26,6.28, 6.29, 6.35, 6.426.28, 6.29, 6.35, 6.42

YMS 6.3YMS 6.3

General Probability RulesGeneral Probability Rules

►Rule for disjoint eventsRule for disjoint events P(one or more of A, B, C) = P(A) + P(B) + P(one or more of A, B, C) = P(A) + P(B) +

P(C)P(C)

►Two events are independent if Two events are independent if P(B|P(B|A)=P(B)A)=P(B)

►Conditional ProbabilityConditional Probability The probability of one event under The probability of one event under

the condition that we know another the condition that we know another eventevent

►General Multiplication Rule (rewrite for General Multiplication Rule (rewrite for conditional probability)conditional probability) P(A and B) = P(A)*P(B|A)P(A and B) = P(A)*P(B|A)

►Bayes’s RuleBayes’s Rule Don’t memorize!Don’t memorize! Use Tree DiagramsUse Tree Diagrams

6.3 Practice – p365 #6.51, 6.52, 6.58, 6.59,6.3 Practice – p365 #6.51, 6.52, 6.58, 6.59,6.61, 6.64, 6.656.61, 6.64, 6.65