Statistics General Probability Rules. Union The union of any collection of events is the event that...

StatisticsStatistics

General Probability RulesGeneral Probability Rules

UnionUnion

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

Addition Rule for Disjoint EventsAddition Rule for Disjoint Events

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

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)

Not DisjointNot Disjoint

If events A and B are not disjoint, If events A and B are not disjoint, they can occur simultaneously.they can occur simultaneously.

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 B P A P B P A B

Deborah and Marshall problemDeborah and Marshall problem

P 362, example 6.17P 362, example 6.17 Deborah = 0.7Deborah = 0.7 Matthew = 0.5Matthew = 0.5 Both = 0.3Both = 0.3

Question – What is the probability that at least one of them is promoted?

What is the probability of at least What is the probability of at least one of them is promoted?one of them is promoted?

P(at least one) = 0.7 + 0.5 – 0.3P(at least one) = 0.7 + 0.5 – 0.3 P(at least one) = 0.9P(at least one) = 0.9

Table of probabilitiesTable of probabilities

PromotePromotedd

Not Not promotedpromoted

totaltotal

DebDeb PromotedPromoted 0.30.3 0.70.7

Not Not promotedpromoted

TotalTotal 0.50.5 1.01.0

QuestionsQuestions P(D and M) = P(D and M) = P(D and not M) = P(D and not M) = P(Not D and M) = P(Not D and M) = P(Not D and not M) = P(Not D and not M) =

AnswersAnswers

P(D and M) = 0.3P(D and M) = 0.3 P(D and not M) = 0.4P(D and not M) = 0.4 P(Not D and M) = 0.2P(Not D and M) = 0.2 P(Not D and not M) = 0.1P(Not D and not M) = 0.1

Problems to doProblems to do

46, 5346, 53

Conditional ProbabilityConditional Probability

the probability of an event the probability of an event happening knowing that another happening knowing that another event has happened.event has happened.

Written as P(AlB) the probability of B Written as P(AlB) the probability of B happening knowing that A has happening knowing that A has happened.happened.

18-2918-29 30-6430-64 65+65+ totaltotal

MarriedMarried 7,8427,842 43,80843,808 8,2708,270 59,92059,920

Never Never MarriedMarried

13,93013,930 7,1847,184 751751 21,86521,865

WidowedWidowed 3636 2,5232,523 8,3858,385 10,94410,944

DivorcedDivorced 704704 9,1749,174 1,2631,263 11,14111,141

TotalTotal 22,51222,512 62,68962,689 18,66918,669 103,870103,870

A = the woman chosen is young, A = the woman chosen is young, ages 18 to 29ages 18 to 29

B = the woman chosen is marriedB = the woman chosen is married

P(A) = 22,512/103,870 = 0.217P(A) = 22,512/103,870 = 0.217 P(A and B) = 7,842/103,870 = 0.075P(A and B) = 7,842/103,870 = 0.075

Probability she is married given that Probability she is married given that she is young.she is young.

P(B l A) = 7,842/22,512 = 0.348P(B l A) = 7,842/22,512 = 0.348

General multiplication rule for two General multiplication rule for two eventsevents

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

Definition of Conditional ProbabilityDefinition of Conditional Probability

( )( | )

( )

P AandBP B A

P A

ProblemsProblems

56, 5856, 58

Extended Multiplication rulesExtended Multiplication rules

Intersection: the intersection of any Intersection: the intersection of any collection of events is the event that collection of events is the event that all of the events occur.all of the events occur.

ExampleExample

The intersection of three events A, B, The intersection of three events A, B, and C has the probabilityand C has the probability

P(A and B and C) = P(A and B and C) = = P(A)P(B|A)P(C|A and B)= P(A)P(B|A)P(C|A and B)

Future of High School AthletesFuture of High School Athletes

Only 5% of male high school Only 5% of male high school basketball, baseball and football basketball, baseball and football players go on to play at the college players go on to play at the college level. Of these, only 1.7% enter level. Of these, only 1.7% enter major league professional sports. major league professional sports. About 40% of the athletes who About 40% of the athletes who compete in college and then reach compete in college and then reach the pros have a career of more than the pros have a career of more than 3 years.3 years.

EventsEvents

A = {competes in college}A = {competes in college} B = {competes professionally}B = {competes professionally} C = {pro career longer than 3 years}C = {pro career longer than 3 years}

P(A) = 0.05P(A) = 0.05 P(B|A) = 0.017P(B|A) = 0.017 P(C|A and B) = 0.4P(C|A and B) = 0.4

P(A and B and C) = P(A and B and C) = = P(A)P(B|A)P(C|A and B)= P(A)P(B|A)P(C|A and B) = 0.05 x 0.017 x 0.40 = 0.05 x 0.017 x 0.40 = 0.00034= 0.00034 Only 3 out of every 10,000 high Only 3 out of every 10,000 high

school athletes can expect to school athletes can expect to compete in college and have a compete in college and have a career greater than 3 yearscareer greater than 3 years

Tree DiagramsTree Diagrams

The probability P(B) is the sum of the probabilities of the two branches ending at B.

Probability of Probability of reaching B given reaching B given college is college is

0.05x0.017=0.000850.05x0.017=0.00085

Probability of reaching Probability of reaching B not going to college B not going to college is is

0.95x0.0001=0.0000950.95x0.0001=0.000095

Probability of P(B) = Probability of P(B) = 0.00085+0.0000950.00085+0.000095

= 0.000945= 0.000945 Or about 9 students out of 10,000 Or about 9 students out of 10,000

will play professional sports.will play professional sports.

ProblemsProblems

64, 67, 70, 77, 79, 80, 83, 8764, 67, 70, 77, 79, 80, 83, 87