probability and simulation

Probability and Probability and Simulation Simulation GENERATING SAMPLE SPACEGENERATING SAMPLE SPACE

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.

•

EXAMPLEEXAMPLE

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 2 P(of having 2

girls)girls)

=2.5=2.5

=.25=.25

=.125=.125

=.50=.50

=.375=.375

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

Being able to properly enumerate Being able to properly enumerate the outcomes in a sample space the outcomes in a sample space will be critical in determining will be critical in determining probabilities. Enumeration and probabilities. Enumeration and tree diagram will be very helpful to tree diagram will be very helpful to eliminate in accidentally eliminate in accidentally overlooking any outcomesoverlooking any outcomes

NOTE!NOTE!

Organize a list of Organize a list of possible outcomes when possible outcomes when you toss a coin 4 timesyou toss a coin 4 times

T T T TT T T T H T T TH T T T H H T TH H T T H H H TH H H T HHHHHHHH

T H T TT H T T H T H TH T H T H H T HH H T H

T T H TT T H T H T T HH T T H H T H HH T H H

T T T HT T T H T H H TT H H T T H H HT H H H

T H T HT H T H

T T H HT T H H

0 heads0 heads 1 head1 head 2 heads2 heads 3 heads3 heads 4 heads4 heads

Multiplication Principle:Multiplication Principle:

If you can do one task in x1 number If you can do one task in x1 number of ways and a second task in x2 of ways and a second task in x2

number of ways, then both can be number of ways, then both can be done in (x1) (x2) number of waysdone in (x1) (x2) number of ways

Example: Example: Sample space of tossing a coin 3 Sample space of tossing a coin 3 times: times: 2 x 2 x 2 = 82 x 2 x 2 = 8

Odds For Meeting A FemaleThe Odds For Meeting A FemaleThe probability of a young man meeting a probability of a young man meeting a desirable and receptive young female desirable and receptive young female increases by exponential progression increases by exponential progression when he is already in the company of: when he is already in the company of:

(1) a date (1) a date (2) his wife (2) his wife (3) a better looking and richer male (3) a better looking and richer male friendfriend

WHICH ONE ARE YOU?WHICH ONE ARE YOU?