A.A. P. STATISTICSP. STATISTICSLESSON 6 – 2 (DAY2)LESSON 6 – 2 (DAY2)

PROBABILITY RULESPROBABILITY RULES

ESSENTIAL QUESTION: What ESSENTIAL QUESTION: What are the probability rules and how are the probability rules and how are they used to solve problems?are they used to solve problems?

Objectives :Objectives : To become familiar with the probability To become familiar with the probability

rules.rules. To use the probability rules to solve To use the probability rules to solve

problems in which they may be used.problems in which they may be used.

Probability RulesProbability Rules Rule 1 : The probability P(A) of any event Rule 1 : The probability P(A) of any event A A

satisfies 0 ≤ P(A) ≤ 1.satisfies 0 ≤ P(A) ≤ 1. Rule 2 : If S is the sample space in Rule 2 : If S is the sample space in

probability model, then P(S) = 1probability model, then P(S) = 1 Rule 3 : The Rule 3 : The compliment compliment of any event A is the event that A of any event A is the event that A

does not occur, written as Adoes not occur, written as Ac c . The . The compliment rule compliment rule states thatstates thatP(AP(Acc) = 1 – P(A)) = 1 – P(A)

Rule 4 : Two events A and B are Rule 4 : Two events A and B are disjoint disjoint ( also called mutually ( also called mutually exclusive ) if they have no outcomes in common and so can exclusive ) if they have no outcomes in common and so can never occur simultaneously. never occur simultaneously.

If A and B are disjoint, If A and B are disjoint,

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

This is the This is the addition rule addition rule for disjoint events.for disjoint events.

Set NotationSet Notation

A U B – read “ A union B” is the set of all A U B – read “ A union B” is the set of all outcomes that are either in A or B’outcomes that are either in A or B’

Empty event – The event that has no outcomes in Empty event – The event that has no outcomes in it. (ø)it. (ø)

If two events A and B are disjoint (mutually If two events A and B are disjoint (mutually exclusively), we can write A ∩ B = ø, read exclusively), we can write A ∩ B = ø, read “intersect B is empty.”“intersect B is empty.”

A picture like that shows the Sample space S as a A picture like that shows the Sample space S as a rectangular area and events as areas within S is rectangular area and events as areas within S is called a called a Venn diagram.Venn diagram.

Venn DiagramVenn Diagram

The events A and B are disjoint because The events A and B are disjoint because they do not overlap;they do not overlap;

AB

S

Compliment ACompliment Acc

The compliment AThe compliment Acc contains exactly the contains exactly the outcomes not in A. Note that we could outcomes not in A. Note that we could write A U Awrite A U Ac c = A ∩ B = ø.= A ∩ B = ø.

A Ac

Example 6.8Example 6.8

Page 344 Page 344

Example 6.9Example 6.9

Page 344 probabilities for rolling dice.Page 344 probabilities for rolling dice.

Assigning probabilities: finite Assigning probabilities: finite number of outcomesnumber of outcomes

PROBABILITIES IN A FINITE SAMPLE PROBABILITIES IN A FINITE SAMPLE SPACESPACE

Assign a probability to each individual Assign a probability to each individual outcome. These probabilities must be outcome. These probabilities must be between numbers 0 and 1 must have sum between numbers 0 and 1 must have sum 1.1.

The probability of any event is the sum of The probability of any event is the sum of probabilities of the outcomes making up probabilities of the outcomes making up the event.the event.

Benford’s LawBenford’s Law

Page 345 example 6.10Page 345 example 6.10

Used in accounting as a test for faked Used in accounting as a test for faked numbers in tax returns, payment records, numbers in tax returns, payment records, invoices, expense account claims, and invoices, expense account claims, and many other settings often display patterns many other settings often display patterns that aren’t present in legitimate records. that aren’t present in legitimate records. Usually applies to first digits.Usually applies to first digits.

Assigning probabilities: equally Assigning probabilities: equally likely outcomeslikely outcomes

Assigning correct probabilities to individual Assigning correct probabilities to individual outcomes often requires long observation outcomes often requires long observation of the random phenomenon. In some of the random phenomenon. In some special circumstances, however, we are special circumstances, however, we are willing to assume that individual outcomes willing to assume that individual outcomes are equally likely because of some are equally likely because of some balance in the phenomenon.balance in the phenomenon.

Equally likely OutcomesEqually likely Outcomes

If a random phenomenon has k possible If a random phenomenon has k possible outcomes, all equally likely, then each outcomes, all equally likely, then each individual outcome has probability 1/k. individual outcome has probability 1/k. The probability of any event A is The probability of any event A is

P(A) = count of outcomes in AP(A) = count of outcomes in A

= count of outcomes in A= count of outcomes in ACount of outcomes in S

k

The Multiplication Rule for The Multiplication Rule for Independent EventsIndependent Events

Rule 5 : Two events A and B are Rule 5 : Two events A and B are independent independent if knowing that one occurs if knowing that one occurs does not change the probability that the does not change the probability that the other occurs. If A and B are independent,other occurs. If A and B are independent,

P(A and B) = P(A)P(B)P(A and B) = P(A)P(B)

This is the This is the multiplication rule multiplication rule for independent for independent events.events.

Venn diagram of independent Venn diagram of independent eventsevents

A

B

A and B

Independent and DisjointIndependent and Disjoint

Disjoint – Mutually exclusiveDisjoint – Mutually exclusive

Independent – The outcome of one trial must not Independent – The outcome of one trial must not influence the outcome of any other.influence the outcome of any other.

Unlike disjointness or compliments, independence Unlike disjointness or compliments, independence cannot be pictured by a Venn diagram, because cannot be pictured by a Venn diagram, because it involves the probabilities of the events rather it involves the probabilities of the events rather than just the outcomes that make up the events.than just the outcomes that make up the events.