AP STATISTICSAP STATISTICSLESSON 6.3LESSON 6.3

(DAY 1)(DAY 1)GENERAL PROBABILITY RULESGENERAL PROBABILITY RULES

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ESSENTIAL QUESTION: ESSENTIAL QUESTION:

What are general probability rules What are general probability rules and how are they used to solve and how are they used to solve probability problems?probability problems?

Objectives: Objectives: To become familiar with general probability To become familiar with general probability

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

problems.problems. To use Venn diagrams, Tree diagrams and To use Venn diagrams, Tree diagrams and

tables to solve probability problems.tables to solve probability problems.

Rules of probabilityRules of probabilityRule 1: 0 ≤ P(A) ≤ 1 for any A.Rule 1: 0 ≤ P(A) ≤ 1 for any A.Rule 2: P(S) = 1Rule 2: P(S) = 1Rule 3: Rule 3: Compliment ruleCompliment rule: : For any event For any event A,A,

P(AP(Acc) = 1 – P(A)) = 1 – P(A)Rule 4: Rule 4: Addition rule: Addition rule: If A and B are If A and B are

disjointdisjoint events, thenevents, thenP(A or B ) = P(A) + P(B)P(A or B ) = P(A) + P(B)

Rule 5: Rule 5: Multiplication ruleMultiplication rule: If : If A and B are A and B are independent independent events, then events, then

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

UnionUnion

The union of any collection of events is the The union of any collection of events is the event that at least one of the collection event that at least one of the collection occurs.occurs.

AB

C

S

The addition rule for disjoint events: P(A or B or C ) = P(A) + P( B) + P(C) when events A, B, and C are disjoint.

Addition rule for disjoint eventsAddition rule for disjoint events

If events A, B, and C are disjoint in the sense If events A, B, and C are disjoint in the sense that no two have any outcomes in common, that no two have any outcomes in common, thenthen

P( one or more of A, B, C ) = P(A) + P(B) + P(C)P( one or more of A, B, C ) = P(A) + P(B) + P(C)

This rule extends to any number of disjoint This rule extends to any number of disjoint events. events.

Example 6.16Example 6.16

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The general addition rule for the union The general addition rule for the union of two events:of two events:

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

AB

A and B

P ( A and B ) is called joint probability.

General addition rule for unions of General addition rule for unions of two events.two events.

For any two events A and B’For any two events A and B’

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

Equivalently,Equivalently,

P(A U B ) = P(A) + P(B) – P( A ∩ B )P(A U B ) = P(A) + P(B) – P( A ∩ B )

Example 6.17Example 6.17

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Conditional ProbabilityConditional Probability

P(A/B) – Conditional probability – gives the P(A/B) – Conditional probability – gives the probability of one event under the probability of one event under the condition that we know another event.condition that we know another event.

Example 6.19Example 6.19

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General Multiplication Rule for any General Multiplication Rule for any Two EventsTwo Events

The probability that both of two events A and B The probability that both of two events A and B happen together can be found by happen together can be found by

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

Here P(B/A) is the conditional probability that B Here P(B/A) is the conditional probability that B occurs given the information that A occurs.occurs given the information that A occurs.

In words, this rule says that for both of two events In words, this rule says that for both of two events to occur, first one must occur and then, given to occur, first one must occur and then, given that the first event has occurred, the second that the first event has occurred, the second must occur.must occur.

Example 6.20Example 6.20

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Definition of conditional probabilityDefinition of conditional probability

When P(A) > 0, the conditional probability of When P(A) > 0, the conditional probability of B given A isB given A is

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

Be sure to keep in mind the distinct roles in Be sure to keep in mind the distinct roles in P(B/A) of the event B whose probability we P(B/A) of the event B whose probability we are computing and the event A that are computing and the event A that represents the information we are given.represents the information we are given.

P (A)

Example 6.21Example 6.21

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