1. Counting Techniques
Prob & Stats

2. Tree Diagrams
When calculating probabilities, you need to know the total number of _____________ in the ______________.
outcomes
sample space

3. On the first flip you could get…..
Tree Diagrams Example H H H T
Use a TREE DIAGRAM to list the sample space of 2 coin flips. T H
Sample Space T T H H
Now you could get…
If you got H T
On the first flip you could get…..
YOU H
Now you could get…
If you got T T T

4. Tree Diagram Example Mr. Milano’s Closet 3 Shirts 2 Pants
2 Pairs of Shoes

5. Dress Mr. Milano
List all of Mr. Milano’s outfits 1 2

6. Dress Mr. Milano
List all of Mr. Milano’s outfits 1 2 3 4

7. Dress Mr. Milano
List all of Mr. Milano’s outfits 1 2 3 4 5 6

8. Dress Mr. Milano
List all of Mr. Milano’s outfits 1 2 3 4 5 6 7 8

9. Dress Mr. Milano
List all of Mr. Milano’s outfits 1 2 3 4 5 6 7 8 9 10

10. Dress Mr. Milano List all of Mr. Milano’s outfits 1 2 3 4 5 6 7 8 9 1011 12

11. 1 6/12 1/2 1/3 4/12 Dress Mr. Milano List all of Mr. Milano’s outfits
If Mr. Milano picks an outfit with his eyes closed…….
List all of Mr. Milano’s outfits 1 2
P(brown shoe) =
6/12 3
1/2 4
P(polo) = 5
1/3
4/12 6 7
P(lookin’ cool) = 1 8 9 10 11 12

12. Multiplication Rule of Counting
The size of the sample space is the ___________ of our probability
So we don’t always need to know what each outcome is, just the of outcomes.
denominator
number

13. Multiplication Rule of Compound Events
If…
X = total number of outcomes for event A
Y = total number of outcomes for event B
Then number of outcomes for A followed by B = _________
x times y

14. Multiplication Rule: Dress Mr. Milano
Mr. Milano had 3 EVENTS 2 2 3
shoes
pants
shirts
How many outcomes are there for EACH EVENT?
2(2)(3) = 12 OUTFITS

15. Permutations
Sometimes we are concerned with how many ways a group of objects can be __________.
arranged
How many ways to arrange books on a shelf
How many ways a group of people can stand in line
How many ways to scramble a word’s letters

16. Example: 3 People, 3 Chairs
Wonder Woman’s invisible plane has 3 chairs.
There are 3 people who need a lift.
How many seating options are there?
Example: 3 People, 3 Chairs
6 Seating Options!
Think of each chair as an EVENT
Batman driving
Wonder Woman driving 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
Superman driving

17. Example: 5 People, 5 Chairs
The batmobile has 5 chairs.
There are 5 people who need a lift.
How many seating options are there?
=120 Seating Options 5 4 3 2 1
Multiply!!
This is a PERMUTATION of 5 objects

18. Commercial Break: FACTORIAL
shown with ! 
Multiply all integers ≤ the number 
0! =
1! =
Calculate 6!
What is 6! / 5!? 5!
5! = 5(4)(3)(2)(1) = 120 1 1
6! = 6(5)(4)(3)(2)(1) = 720

19. Commercial Break: FACTORIAL
denoted with ! 
Multiply all integers ≤ the number 
0! =
1! =
Calculate 6!
What is 6! / 5!? 5! 1 1 =6
6(5)(4)(3)(2)(1)
5(4)(3)(2)(1)

20. Example: 5 People, 5 Chairs
The batmobile has 5 chairs.
There are 5 people who need a lift.
How many seating options are there? 5!
=120 Seating Options 5 4 3 2 1
Multiply!!
This is a PERMUTATION of 5 objects

21. Permutations: Not everyone gets a seat!
It’s time for annual Justice League softball game.
How many ways could your assign people to play 1st, 2nd, and 3rd base?
What if I choose these 3?
You have to choose 3 AND arrange them
Think of the possibilities!

22. Permutations: Not everyone gets a seat!
It’s time for annual Justice League softball game.
How many ways could your assign people to play 1st, 2nd, and 3rd base?
What if I choose these 3?
You have to choose 3 AND arrange them
Think of the possibilities!

23. Permutations: Not everyone gets a seat!
It’s time for annual Justice League softball game.
How many ways could your assign people to play 1st, 2nd, and 3rd base?
What if I choose these 3?
You have to choose 3 AND arrange them
Think of the possibilities!

24. Permutations: Not everyone gets a seat!
It’s time for annual Justice League softball game.
How many ways could your assign people to play 1st, 2nd, and 3rd base?
BUT… What if I choose THESE 3?
You have to choose 3 AND arrange them
Think of the possibilities!

25. Permutations: Not everyone gets a seat!
It’s time for annual Justice League softball game.
How many ways could your assign people to play 1st, 2nd, and 3rd base?
BUT… What if I choose THESE 3?
You have to choose 3 AND arrange them
Think of the possibilities!

26. Permutations: Not everyone gets a seat!
It’s time for annual Justice League softball game.
How many ways could your assign people to play 1st, 2nd, and 3rd base?
BUT… What if I choose THESE 3?
You have to choose 3 AND arrange them
Think of the possibilities!

27. Permutations: Not everyone gets a seat!
It’s time for annual Justice League softball game.
How many ways could your assign people to play 1st, 2nd, and 3rd base?
BUT… What if I choose THESE 3?
This is going to take FOREVER
You have to choose 3 AND arrange them
Think of the possibilities!

28. 5 You have 3 EVENTS? How many outcomes for each event
How many outcomes for this event! 5
You have to choose 3 AND arrange them

29. 4 5 You have 3 EVENTS? You have to choose 3 AND arrange them
How many outcomes for this event! 4
Now someone is on FIRST 5
You have to choose 3 AND arrange them

30. 4 3 5 You have 3 EVENTS? 5(4)(3) = 120 POSSIBLITIES
And on SECOND 4
Now someone is on FIRST 3 5
How many outcomes for this event!
You have to choose 3 AND arrange them

31. Permutation Formula You have You select
This is the number of ways you could select and arrange in order:
n objects
r objects
Another common notation for a permutation is nPr

32. Softball Permutation Revisited
5(4)(3)(2)(1) 5! 5
n =
people to choose from
r = 3
2(1) 2!
(5 – 3)!
spots to fill
5(4)(3) = 120 POSSIBLITIES
You have to choose 3 AND arrange them

33. Combinations
Sometimes, we are only concerned with a group and in which they are selected.
A gives the number of ways to of r objects from a group of size n.
selecting
not the order
combination
select a sample

34. Combination: Duty Calls
There is an evil monster threatening the city.
The mayor calls the Justice League.
He requests that 3 members be sent to combat the menace.
The Justice League draws 3 names out of a hat to decide.
Does it matter who is selected first?
Does it matter who is selected last?
NOPE
NOPE

35. Combination: Duty Calls
Let’s look at the drawing possibilities
These are all the SAME: The monster doesn’t care who got drawn first.
All these outcomes = same people pounding his face
These are all the SAME: The monster doesn’t care who got drawn first.
All these outcomes = same people pounding his face
STOP! This is a waste of time
We’ll count them as ONE OUTCOME
We’ll count them as ONE OUTCOME

36. Combination: Duty Calls
Okay, let’s consider other outcomes
10 Possible Outcomes!

37. Combinations are also denoted nCr
Combination Formula
You have
You want a group of
You what order they are selected in
n objects
r objects
DON’T CARE
Combinations are also denoted nCr
Read “n choose r”

38. Duty Calls: Revisited 5! 3!(5 - 3)! 3!(2)! 5(4)(3)(2)(1) 3(2)(1)(2)(1)
n =
people to choose from
ORDER DOESN’T MATTER
r = 3
spots to fill
5(4)(3)(2)(1) 5! 20 2
3!(5 - 3)!
3!(2)!
3(2)(1)(2)(1)
10 Possible Outcomes!
Now we can go save the city

40. Permutation vs. Combination
Order matters 
Order doesn’t matter 
Permutation
Combination