Introduction to Introduction to ProbabilityProbability

Ginger Holmes Rowell, Ginger Holmes Rowell, Middle TN State UniversityMiddle TN State University

MSP WorkshopMSP WorkshopJune 2007June 2007

ObjectivesObjectives• Understand the concepts ofUnderstand the concepts of

• sample space and probability distribution and sample space and probability distribution and construct sample spaces and distributions in simple construct sample spaces and distributions in simple casescases

• conditional probability and independent events; conditional probability and independent events; understand how to compute the probability of a understand how to compute the probability of a compound eventcompound event

• Use simulations to construct empirical probability Use simulations to construct empirical probability distributions and to make informal inferences about the distributions and to make informal inferences about the theoretical probability distribution theoretical probability distribution

Probability ReviewProbability Review

• DefinitionsDefinitions

• Classical ProbabilityClassical Probability

• Relative Frequency ProbabilityRelative Frequency Probability

• Probability Fundamentals andProbability Fundamentals and Probability Rules Probability Rules

What is Probability?What is Probability?

• ProbabilityProbability the study of chance associated the study of chance associated

with the occurrence of eventswith the occurrence of events

• Types of ProbabilityTypes of Probability• Classical (Theoretical)Classical (Theoretical)• Relative Frequency (Experimental)Relative Frequency (Experimental)

Classical ProbabilityClassical Probability

Rolling dice and tossing a coin are Rolling dice and tossing a coin are activities associated with a activities associated with a classical approach to probability.  classical approach to probability.  In these cases, you can list all the In these cases, you can list all the possible outcomes of an possible outcomes of an experiment and determine the experiment and determine the actual probabilities of each actual probabilities of each outcome. outcome.

Listing Listing All Possible OutcomesAll Possible Outcomes of a Probabilistic Experiment of a Probabilistic Experiment

• There are various ways to list all There are various ways to list all possible outcomes of an possible outcomes of an experiment experiment • EnumerationEnumeration• Tree diagramsTree diagrams• Additional methods – counting Additional methods – counting

fundamentalsfundamentals

Three Children ExampleThree Children Example

• A couple wants to have exactly 3 A couple wants to have exactly 3 children.  Assume that each children.  Assume that each child is either a boy or a girl and child is either a boy or a girl and that each is a single birth.  that each is a single birth. 

• List all possible orderings for List all possible orderings for the 3 children.the 3 children.

EnumerationEnumeration11stst Child Child 22ndnd Child Child 33rdrd Child Child

EnumerationEnumeration11stst Child Child 22ndnd Child Child 33rdrd Child Child

BB BB BB

GG BB BB

BB GG BB

BB BB GG

GG GG BB

GG BB GG

BB GG GG

GG GG GG

Tree DiagramsTree Diagrams

1st Child 2nd Child 3rd Child BBBBBB

BB

G

B

GB

G

BBGBBGBGBBGBBGGBGGGBBGBBGBGGBGGGBGGBGGGGGG

GB

G

B

GB

G

DefinitionsDefinitions

• Sample SpaceSample Space - the list of all - the list of all possible outcomes from a possible outcomes from a probabilistic experiment.  probabilistic experiment.  • 3-Children Example:3-Children Example:

S = {BBB, BBG, BGB, BGG, S = {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG} GBB, GBG, GGB, GGG}

• Each individual item in the list is called Each individual item in the list is called a a Simple EventSimple Event or or Single Event.Single Event.

Probability NotationProbability Notation

P(P(eventevent) = Probability of the ) = Probability of the eventevent occurringoccurring

Example: P(Boy) = P(B) = ½Example: P(Boy) = P(B) = ½

Probability of Single Events Probability of Single Events with Equally Likely Outcomeswith Equally Likely Outcomes       

• If each outcome in the sample space If each outcome in the sample space is is equally likelyequally likely, then , then the probability of the probability of any one outcome is 1 divided by the any one outcome is 1 divided by the total number of outcomestotal number of outcomes..    

outcomes ofnumber total

1event) simple(

outcomes,likely equally For

P

Three Children Example Three Children Example ContinuedContinued

• A couple wants 3 children. A couple wants 3 children. Assume the chance of a boy or Assume the chance of a boy or girl is girl is equally likelyequally likely at each birth.  at each birth. 

• What is the What is the probabilityprobability that they that they will have will have exactly 3 girlsexactly 3 girls? ? 

• What is the What is the probabilityprobability of ofhaving having exactly 3 boysexactly 3 boys??

Probability of Combinations of Probability of Combinations of Single EventsSingle Events

• An An EventEvent can be a combination can be a combination of of Single EventsSingle Events..

• The probability of such an event The probability of such an event is the sum of the individual is the sum of the individual probabilities.probabilities.

Three Children Example Three Children Example ContinuedContinued

P(exactly 2 girls) = __P(exactly 2 girls) = __

P(exactly 2 boys) = __P(exactly 2 boys) = __

P(at least 2 boys) = __P(at least 2 boys) = __

P(at most 2 boys) = __P(at most 2 boys) = __

P(at least 1 girl) = __P(at least 1 girl) = __

P(at most 1 girl) = __P(at most 1 girl) = __

• Sample Sample space =space =

Types of ProbabilityTypes of Probability

• Classical (Theoretical)Classical (Theoretical)

• Relative Frequency Relative Frequency (Experimental, Empirical)(Experimental, Empirical)

Relative Frequency ProbabilityRelative Frequency Probability

• Uses actual experience to Uses actual experience to determine the likelihood of an determine the likelihood of an outcome.outcome.

• What isWhat isthe chancethe chanceof makingof makinga B or better?a B or better?

GradeGrade FrequencyFrequency

AA 2020

BB 3030

CC 4040

Below CBelow C 1010

Relative Frequency Probability Relative Frequency Probability is Great Fun for Teachingis Great Fun for Teaching

• Rolling DiceRolling Dice

• Flipping CoinsFlipping Coins

• Drawing from Bags without Looking Drawing from Bags without Looking (i.e. Sampling)(i.e. Sampling)

• Sampling with M&M's  Sampling with M&M's  ((http://mms.com/cai/mms/faq.html#whhttp://mms.com/cai/mms/faq.html#what_percentat_percent))

Empirical ProbabilityEmpirical Probability

• Given a frequency distribution, Given a frequency distribution, the probability of an event, E, the probability of an event, E, being in a given group isbeing in a given group is

n

xP

ondistributi in the sfrequencie total

group theoffrequency E)(

Two-way Tables and Two-way Tables and ProbabilityProbability

• FindFind

P(M) P(M)

P(A)P(A)

P(A and M)P(A and M)

Made Made AA

Made Made

< A< ATotalTotal

MaleMale 3030 4545

FemaleFemale 6060 6565

TotalTotal

Teaching IdeaTeaching Idea

• Question: How Can You Win at Question: How Can You Win at Wheel of Fortune?Wheel of Fortune?

• Answer: Use Relative Frequency Answer: Use Relative Frequency Probability (see handout)Probability (see handout)

Source. Krulik and Rudnick. “Teaching Middle School Mathematics Activities, Materials and Problems.” p. 161.  Allyn & Bacon, Boston. 2000.

Probability FundamentalsProbability Fundamentals

• What is What is wrongwrong with the statements? with the statements?• The probability of rain today is -10%.The probability of rain today is -10%.• The probability of rain today is 120%.The probability of rain today is 120%.• The probability of rain or no rain today is The probability of rain or no rain today is

90%.90%.

1) (

1)(

0)(

spacesampleP

eventP

eventP

Probability RulesProbability Rules

Let A and B be eventsLet A and B be events

Complement Rule:Complement Rule:

P(A) + P(not A) = 1P(A) + P(not A) = 1

Set NotationSet Notation

Union: A or B Union: A or B (inclusive “or”)(inclusive “or”)

BA

BA

Intersection: A and BIntersection: A and B

Probability RulesProbability Rules

Union P(AUB) = P(A or B)Union P(AUB) = P(A or B)

)()()()( BAPBPAPBAP

• Venn DiagramsVenn Diagrams

• Kyle Siegrist’s Venn Diagram Kyle Siegrist’s Venn Diagram AppletApplet

http://www.math.uah.edu/stat/applhttp://www.math.uah.edu/stat/applets/index.xmlets/index.xml

Teaching IdeaTeaching Idea

Two-way Tables and Two-way Tables and ProbabilityProbability

• FindFind

P(M)P(M)

P(A)P(A)

P(A and M)P(A and M)

P(A if M)P(A if M)

Made Made AA

Made Made

< A< A

TotalTotal

MaleMale 3030 4545 7575

FemaleFemale 6060 6565 125125

TotalTotal 9090 110110 200200

Conditional ProbabilityConditional Probability

P(A|B) = the conditional P(A|B) = the conditional probability of event A happening probability of event A happening given that event B has happenedgiven that event B has happened

“ “probability of A given B”probability of A given B”

)(

)()|(

BP

BAPBAP

IndependenceIndependence

• Events A and B are Events A and B are “Independent” if and only if“Independent” if and only if

)()|( APBAP

• Using the data in the two-way Using the data in the two-way table, is making an “A” table, is making an “A” independent from being male? independent from being male?

HomeworkHomework