Probability and Probability and Simulation Simulation The Study of RandomnessThe Study of Randomness

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

• EnumerationEnumeration

• Tree diagramsTree diagrams

• Additional methods – counting Additional methods – counting

fundamentalsfundamentals

Three Children ExampleThree Children Example• A couple wants to have exactly A couple wants to have exactly 3 children.  Assume that each 3 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 the List all possible orderings for the 3 children.3 children.

•

HomeworkHomework

EnumerationEnumeration

11stst 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

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

Tree DiagramsTree Diagrams

1st Child __ 2nd Child __ 3rd Child

BBBBBBB

G

B

GB

G

BBGBBGBGBBGB

BGGBGGGBBGBB

GBGGBGGGBGGB

GGGGGG

B

GB

G

B

GB

G

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

PRACTICE:PRACTICE:Using the previous Using the previous conditions, what is the probability conditions, what is the probability of these events from happening?of these events from happening?

P(of getting 2 consecutive boys)P(of getting 2 consecutive boys)

P(of getting a boy then a girl)P(of getting a boy then a girl)

P(of getting all boys)P(of getting all boys)P(of having a girl as a third P(of having a girl as a third

child)child)P(of having exactly 2 P(of having exactly 2

girls)girls)

=3/8=3/8

=4/8 or 1/2=4/8 or 1/2

=1/8=1/8

=4/8 or 1/2=4/8 or 1/2

=3/8=3/8

A glass jar contains A glass jar contains 6 red6 red, , 5 green5 green, , 8 blue8 blue and and 3 yellow 3 yellow marbles. If a single marble is marbles. If a single marble is chosen at random from the jar, what is the chosen at random from the jar, what is the probability of choosing a red marble? a green probability of choosing a red marble? a green marble? a blue marble? a yellow marble? marble? a blue marble? a yellow marble? P[red] P[red]

= .27= .27P[green] P[green] = .23 = .23P[blue] P[blue] = .36= .36P[yellow] P[yellow] = .14= .14

Warm-upWarm-up

Compliment of a Compliment of a probabilityprobabilityThe chances of raining The chances of raining

today is 25%. What is the today is 25%. What is the probability that itprobability that it’’s NOT s NOT

going to rain today?going to rain today?

P(rain) = 25%P(rain) = 25% PP ’’(rain) = 75%(rain) = 75%