Tree Diagram &
Counting Methods
Prob & Stats
Tree Diagrams

When calculating probabilities,
you need to know the total
outcomes
number of _____________
in
the ______________.
sample space
Tree Diagrams Example

Use a TREE DIAGRAM to list the
Sample
sample space of 2 coin flips.
Space
H
H
you could
got H get…
NowIf you
T
YOU
On the first flip you
could get…..
H
NowIfyou
youcould
got Tget…
T
T
A tree diagram is a way of
describing all the
possible outcomes from
a series of events
A tree diagram is a way of calculating
the probability of all the possible
outcomes from a series of events
A list of possible events could include
• Picking a square on a chess board
Black or white
• Flipping a coin
Head or tail
• Taking a sweet from a jar
of red, yellow and green
sweets
What is
There
different
are 3
about
outcomes
the outcome
here? or green
Red, yellow
All these events have definite outcomes
CLICK
Each event can be represented by a tree diagram
Picking a sweet from a jar containing red, yellow and green sweets
Picking a square on a chess board
Flipping a coin
REMEMBER – there are 3 outcomes
BLACK
HEAD
RED
WHITE
TAIL
YELLOW
The diagram shows you have a choice of 2 paths (branches)
The diagram shows you
GREEN
have a choice of 2 paths
(branches)
The diagram shows you have a choice of 3 paths (branches)
CLICK
The list of possible events and their outcomes are
Probability of black = 0.5
• Picking a square on a
chess board
Probability of white = 0.5
Black or white
Equal number of black
and white
squares
Probability
of head
= 0.5
• Flipping a coin
Therefore
Equalofchance
probability
of head or
Probability
tailthe
= 0.5
Head
or
tail
of
tails
each outcome is
• Picking a sweet from
a jar of red, yellow and
green sweets
0.5
Therefore the probability
of each
outcome
is
The
remaining
event
needs
more or green
Red,
0.5 yellow
investigation
All these outcomes have definite probabilities
CLICK
EVENT : Picking a sweet from a jar of coloured
sweets
In the jar there are
12 red sweets
5 yellow sweets
8 green sweets
Probability of picking a red is 12 out of 25
0.48
Probability of picking a yellow is
0.20
Probability of picking a green is
0.32
CLICK
The tree diagrams can now be completed
Picking a square on a chess board
Probability of BLACK = 0.5
0.5
Probability of WHITE = 0.5 0.5
BLACK
WHITE
0.5
0.5
Now we add the probabilities
THE TREE DIAGRAM IS NOW COMPLETE
CLICK
Similarly, the tree diagram for flipping a coin is
Picking a sweet from a jar of red,yellow and green sweets
Probability of RED = 0.48
RED
0.48
Probability of
yellow
0.2
YELLOW
= 0.2
0.32
Probability of GREEN = 0.32
These are the outcomes
The tree diagram is compete
We must add the probabilities
GREEN
Tree Diagram Example

Mr. Arnold’s Closet
3 Shirts
2 Pairs of Shoes
2 Pants
Dress Mr. Arnold

List all of Mr. Arnold’s outfits
1
2
Dress Mr. Arnold

List all of Mr. Arnold’s outfits
1
2
3
4
Dress Mr. Arnold

List all of Mr. Arnold’s outfits
1
2
3
4
5
6
Dress Mr. Arnold

List all of Mr. Arnold’s outfits
1
2
3
4
5
6
7
8
Dress Mr. Arnold

List all of Mr. Arnold’s outfits
1
2
3
4
5
6
7
8
9
10
Dress Mr. Arnold

List all of Mr. Arnold’s outfits
1
2
3
4
5
6
7
8
9
10
11
12
Dress Mr. Arnold

List all of Mr. Arnold’s outfits
If Mr. Arnold picks an
outfit with his eyes
closed…….
1
2
P(brown shoe) =
3
6/12
1/2
4
P(polo) =
5
1/3
4/12
6
7
8
9
10
11
12
P(lookin’ cool) =
1
Multiplication Rule of Counting


The size of the sample space is
denominator of our
the ___________
probability
So we don’t always need to
know what each outcome is, just
the number of outcomes.
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
times y
followed by B = x_______
= (x )(y)
Multiplication Rule:
Dress Mr. Arnold

Mr. Reed had 3 EVENTS
2
shoes
2
pants
3
shirts
How many outcomes are there
for EACH
EVENT?
2(2)(3)
= 12
OUTFITS
Permutations

Sometimes we are concerned
with how many ways a group of
arranged
objects can be __________.
•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
Example:
3 People, 3 Chairs
Hercules driving



We has 3 chairs.
There are 3 people who need a lift.
How many seating options are there?
of each chair as
6 Think
Seating
Options!
an EVENT
3
2
1
How
many
ways
the
Now
Now
that
thethe
first1st2 could
is
are
filled?
filled.
st chair be filled? ndrd
1many
3(2)(1)
= options
6ways
OPTIONS
How
How
many
to
forfill2 3 ??
Superman driving
Batman driving
Example:
5 People, 5 Chairs
5
4
3



The batmobile has 5 chairs.
There are 5 people who need a lift.
How many seating options are there?
2
1
=120
Seating Options
Multiply!!
This is a PERMUTATION of 5 objects
Commercial Break:
FACTORIAL


denoted with !  5!
Multiply all integers ≤ the number 

5! = 5(4)(3)(2)(1) = 120
0! = 1
1! = 1

Calculate 6!

6! = 6(5)(4)(3)(2)(1) = 720

What is 6! / 5!?
Commercial Break:
FACTORIAL


denoted with !  5!
Multiply all integers ≤ the number 

0! = 1
1! = 1
Calculate 6!

What is 6! / 5!?


6(5)(4)(3)(2)(1)
5(4)(3)(2)(1)
=6
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!
5
4
3
2
1
=120
Seating Options
Multiply!!
This is a PERMUTATION of 5 objects
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!
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!
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!
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!
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!
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!
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!
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
You have 3 EVENTS?
How many
outcomes for this
event!
4
Now someone is
on FIRST
5
You have to choose 3 AND
arrange them
You have 3 EVENTS?
5(4)(3) = 60 POSSIBLITIES
And on SECOND
4
Now someone is
on FIRST
3
How many
outcomes for this
event!
You have to choose 3 AND
arrange them
5
Permutation Formula



You have n objects
You select r objects
This is the number of ways you
could select and arrange in
order:
n!
P 
(n  r )!
n
r
Another common notation for a permutation is nPr
Softball Permutation Revisited
n
5!!
5(4)(3)(2)(1)
n
people
to
choose
from
5
n=
Pr 
2!3)!
r )!
r = 3 spots to fill ((5n –2(1)
5(4)(3) = 60 POSSIBLITIES
You have to choose 3 AND
arrange them
Combinations


Sometimes, we are only
concerned with selecting a group
and not the order in which they
are selected.
A combination gives the number
of ways to select a sample of r
objects from a group of size n.
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?
NOPE
Does it matter who is selected last?
NOPE
Combination: Duty Calls
Let’s look at the drawing possibilities
STOP!
This is a waste of
time
These
are all
the SAME:
We’ll
count
them
as
ONE OUTCOME
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.
We’ll count them as
ONE OUTCOME
All these outcomes = same people
pounding his face
Combination: Duty Calls
Okay, let’s consider other outcomes
10 Possible Outcomes!
Combination Formula



You have n objects
You want a group of r objects
You DON’T CARE what order
they are selected in
n!
C 
r!(n  r )!
n
r
Combinations are also denoted nCr
Read “n choose r”
Duty Calls: Revisited
to choose from
n = 5 people
ORDER DOESN’T MATTER
r = 3 spots to fill
C 
n
r
5(4)(3)(2)(1)
n!
5!
3(2)(1)(2)(1)
3!(5
r!(3!(2)!
n -3)!
r )!
10 Possible Outcomes!
Now we can go save the city
20
2
Permutation vs. Combination


Order matters  Permutation
Order doesn’t matter Combination