4.3 Counting 4.3 Counting Techniques Techniques Prob & Stats Prob & Stats
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