4.3 counting techniques prob & stats tree diagrams when calculating probabilities, you need to...

4.3 Counting 4.3 Counting TechniquesTechniques

Prob & StatsProb & Stats

Tree DiagramsTree Diagrams

When calculating probabilities, When calculating probabilities, you need to know the total you need to know the total number of _____________ in number of _____________ in the ______________.the ______________.

outcomessample space

Tree Diagrams ExampleTree Diagrams Example Use a TREE DIAGRAM to list the Use a TREE DIAGRAM to list the

sample space of 2 coin flips.sample space of 2 coin flips.

YOU

On the first flip you could get…..

H

T

If you got HNow you could get…

If you got TNow you could get…

H

T

H

T

SampleSpace

Tree Diagram ExampleTree Diagram Example Mr. Arnold’s ClosetMr. Arnold’s Closet

3 Shirts 2 Pants

2 Pairs of Shoes

Dress Mr. ArnoldDress Mr. Arnold List all of Mr. Arnold’s outfitsList all of Mr. Arnold’s outfits 1

2

Dress Mr. ArnoldDress Mr. Arnold List all of Mr. Arnold’s outfitsList all of Mr. Arnold’s outfits 1

2

3

4

Dress Mr. ArnoldDress Mr. Arnold List all of Mr. Arnold’s outfitsList all of Mr. Arnold’s outfits 1

2

3

4

5

6

Dress Mr. ArnoldDress Mr. Arnold List all of Mr. Arnold’s outfitsList all of Mr. Arnold’s outfits 1

2

3

4

5

6

7

8

Dress Mr. ArnoldDress Mr. Arnold List all of Mr. Arnold’s outfitsList all of Mr. Arnold’s outfits 1

2

3

4

5

6

7

8

9

10

Dress Mr. ArnoldDress Mr. Arnold List all of Mr. Arnold’s outfitsList all of Mr. Arnold’s outfits 1

2

3

4

5

6

7

8

9

10

11

12

Dress Mr. ArnoldDress Mr. Arnold List all of Mr. Arnold’s outfitsList all of Mr. Arnold’s outfits 1

2

3

4

5

6

7

8

9

10

11

12

If Mr. Arnold picks an outfit with his eyes

closed…….

P(brown shoe) =

6/121/2P(polo) =

4/121/3P(lookin’ cool) =

1

Multiplication Rule of CountingMultiplication Rule of Counting

The size of the sample space is The size of the sample space is the ___________ of our the ___________ of our probabilityprobability

So we don’t always need to So we don’t always need to know what each outcome is, just know what each outcome is, just the the of outcomes. of outcomes.

denominator

number

Multiplication Rule of Multiplication Rule of Compound EventsCompound Events

If…If… X = X = total number of outcomes total number of outcomes

for event Afor event A Y = Y = total number of outcomes total number of outcomes

for event Bfor event B Then number of outcomes for A Then number of outcomes for A

followed by B = ____followed by B = ______________ x times y

Multiplication Rule:Multiplication Rule:Dress Mr. ArnoldDress Mr. Arnold

Mr. Reed had 3 EVENTSMr. Reed had 3 EVENTS

pantsshoes shirts

How many outcomes are there for EACH EVENT?

2 2 3

2(2)(3) = 12 OUTFITS

PermutationsPermutations

Sometimes we are concerned Sometimes we are concerned with how many ways a group of with how many ways a group of objects can be __________.objects can be __________.arranged

•How many ways to arrange books on a How many ways to arrange books on a shelfshelf

•How many ways a group of people can How many ways a group of people can stand stand in line in line

•How many ways to scramble a word’s How many ways to scramble a word’s lettersletters

Wonder Woman’s invisible plane has 3 Wonder Woman’s invisible plane has 3 chairs.chairs.

There are 3 people who need a lift.There are 3 people who need a lift. How many seating options are there?How many seating options are there?

Example: Example: 3 People, 3 Chairs3 People, 3 Chairs

Superman driving

Batman drivingWonder Woman driving 6 Seating Options!Think of each chair as

an EVENT

3 2 1

How many ways could the 1st chair be filled?

Now that the 1st is filled?How many options for 2nd?

Now the first 2 are filled.How many ways to fill 3rd?3(2)(1) = 6 OPTIONS

Example: Example: 5 People, 5 Chairs5 People, 5 Chairs

The batmobile has 5 chairs.The batmobile has 5 chairs. There are 5 people who need a lift.There are 5 people who need a lift. How many seating options are there?How many seating options are there?

5 4 3 2 1

Multiply!!

=120Seating Options

This is a PERMUTATION of 5 objects

Commercial Break:Commercial Break:FACTORIALFACTORIAL

denoted with ! denoted with ! Multiply all integers ≤ the number Multiply all integers ≤ the number

0! = 0! = 1! = 1! = Calculate 6! Calculate 6!

What is 6! / 5!? What is 6! / 5!?

5!

5! = 5(4)(3)(2)(1) = 12011

6! = 6(5)(4)(3)(2)(1) = 720

Commercial Break:Commercial Break:FACTORIALFACTORIAL

denoted with ! denoted with ! Multiply all integers ≤ the number Multiply all integers ≤ the number

0! = 0! = 1! = 1! = Calculate 6! Calculate 6!

What is 6! / 5!? What is 6! / 5!?

5!

5(4)(3)(2)(1)

11

6(5)(4)(3)(2)(1) =6

Example: Example: 5 People, 5 Chairs5 People, 5 Chairs

The batmobile has 5 chairs.The batmobile has 5 chairs. There are 5 people who need a lift.There are 5 people who need a lift. How many seating options are there?How many seating options are there?

5 4 3 2 1

Multiply!!

=120Seating Options

This is a PERMUTATION of 5 objects

5!

Permutations:Permutations:Not everyone gets a seat!Not everyone gets a seat! It’s time for annual Justice League softball game.It’s time for annual Justice League softball game. How many ways could your assign people to play 1How many ways could your assign people to play 1stst, 2, 2ndnd, ,

and 3and 3rdrd base? base?

You have to choose 3 AND arrange them

What if I choose these

3?

Think of the possibilities!

Permutations:Permutations:Not everyone gets a seat!Not everyone gets a seat! It’s time for annual Justice League softball game.It’s time for annual Justice League softball game. How many ways could your assign people to play 1How many ways could your assign people to play 1stst, 2, 2ndnd, ,

and 3and 3rdrd base? base?

You have to choose 3 AND arrange them

What if I choose these

3?

Think of the possibilities!

Permutations:Permutations:Not everyone gets a seat!Not everyone gets a seat! It’s time for annual Justice League softball game.It’s time for annual Justice League softball game. How many ways could your assign people to play 1How many ways could your assign people to play 1stst, 2, 2ndnd, ,

and 3and 3rdrd base? base?

You have to choose 3 AND arrange them

What if I choose these

3?

Think of the possibilities!

Permutations:Permutations:Not everyone gets a seat!Not everyone gets a seat! It’s time for annual Justice League softball game.It’s time for annual Justice League softball game. How many ways could your assign people to play 1How many ways could your assign people to play 1stst, 2, 2ndnd, ,

and 3and 3rdrd base? base?

You have to choose 3 AND arrange them

BUT…What if I choose

THESE 3?

Think of the possibilities!

Permutations:Permutations:Not everyone gets a seat!Not everyone gets a seat! It’s time for annual Justice League softball game.It’s time for annual Justice League softball game. How many ways could your assign people to play 1How many ways could your assign people to play 1stst, 2, 2ndnd, ,

and 3and 3rdrd base? base?

You have to choose 3 AND arrange them

BUT…What if I choose

THESE 3?

Think of the possibilities!

Permutations:Permutations:Not everyone gets a seat!Not everyone gets a seat! It’s time for annual Justice League softball game.It’s time for annual Justice League softball game. How many ways could your assign people to play 1How many ways could your assign people to play 1stst, 2, 2ndnd, ,

and 3and 3rdrd base? base?

You have to choose 3 AND arrange them

BUT…What if I choose

THESE 3?

Think of the possibilities!

Permutations:Permutations:Not everyone gets a seat!Not everyone gets a seat! It’s time for annual Justice League softball game.It’s time for annual Justice League softball game. How many ways could your assign people to play 1How many ways could your assign people to play 1stst, 2, 2ndnd, ,

and 3and 3rdrd base? base?

You have to choose 3 AND arrange them

BUT…What if I choose

THESE 3?

Think of the possibilities!

This is going to take

FOREVER

You have 3 EVENTS?You have 3 EVENTS? How many outcomes for each eventHow many outcomes for each event

You have to choose 3 AND arrange them

How many outcomes for this

event!

5

You have 3 EVENTS?You have 3 EVENTS?

You have to choose 3 AND arrange them

How many outcomes for this

event!

4Now someone is

on FIRST

5

You have 3 EVENTS?You have 3 EVENTS?

You have to choose 3 AND arrange them

And on SECOND

4Now someone is

on FIRST

53How many

outcomes for this event!

5(4)(3) = 120 POSSIBLITIES

Permutation FormulaPermutation Formula

You have You have You selectYou select This is the number of ways you This is the number of ways you

could could selectselect and and arrangearrange in in order: order:

)!(

!

rn

nP nr

Another common notation for a permutation is nPr

n objectsr objects

You have to choose 3 AND arrange them

n =r =

5 people to choose from

3 spots to fill )!(

!

rn

nP nr

5!Softball Permutation RevisitedSoftball Permutation Revisited

(5 – 3)!

5(4)(3)(2)(1)

2!2(1)

5(4)(3) = 120 POSSIBLITIES

CombinationsCombinations

Sometimes, we are only Sometimes, we are only concerned with concerned with a group a group and and in which they in which they are selected.are selected.

A A gives the number gives the number of ways to of ways to of of rr objects from a group of size objects from a group of size nn. .

selectingnot the order

combinationselect a sample

CCombination: Duty Callsombination: Duty Calls There is an evil monster threatening There is an evil monster threatening

the city.the city. The mayor calls the Justice League.The mayor calls the Justice League. He requests that He requests that 33 members be sent members be sent

to combat the menace.to combat the menace. The Justice League draws 3 names The Justice League draws 3 names

out of a hat to decide.out of a hat to decide. Does it matter who is selected first?Does it matter who is selected first?

Does it matter who is selected last?Does it matter who is selected last?

NOPE

NOPE

CCombination: Duty Callsombination: Duty CallsLet’s look at the drawing possibilities

STOP!This is a waste of

time

These are all the SAME:The monster doesn’t care who got drawn

first.

All these outcomes = same people pounding his face

We’ll count them as ONE OUTCOME

These are all the SAME:The monster doesn’t care who got drawn

first.

All these outcomes = same people pounding his face

We’ll count them as ONE OUTCOME

CCombination: Duty Callsombination: Duty Calls

Okay, let’s consider other outcomes

10 Possible Outcomes!

Combination FormulaCombination Formula

You have You have You want a group of You want a group of You You what order what order

they are selected inthey are selected in

n objectsr objects

DON’T CARE

)!(!

!

rnr

nC nr

Combinations are also denoted nCr

Read “n choose r”

DDuty Calls: Revisiteduty Calls: Revisited

)!(!

!

rnr

nC nr

n =r =

5 people to choose from

3 spots to fillORDER DOESN’T MATTER

5!

3!(5 - 3)!nrC

3!(2)!

5(4)(3)(2)(1)

3(2)(1)(2)(1)

20

2

10 Possible Outcomes!

Now we can go save the city

Permutation vs. CombinationPermutation vs. Combination

Order matters Order matters Order doesn’t matter Order doesn’t matter

Permutation

Combination