tree diagrams with solved examples

1. COUNTING TECHNIQUE:
Tree Diagrams
BY UNSA SHAKIR

2. Example : Using the Fundamental Counting Principle
A password for a site consists of 4 digits followed
by 2 letters. The letters A and Z are not used, and
each digit or letter many be used more than once.
How many unique passwords are possible?
digit digit digit digit letter letter
10  10  10  10  24  24 = 5,760,000
There are 5,760,000 possible passwords.

3. Example
A password is 4 letters followed by 1 digit.
Uppercase letters (A) and lowercase letters (a) may
be used and are considered different. How many
passwords are possible?
Since both upper and lower case letters can be used,
there are 52 possible letter choices.
letter letter letter letter number
52  52  52  52  10 = 73,116,160
There are 73,116,160 possible passwords.

4. When deciding whether to use permutations or
combinations, first decide whether order is important. Use a
permutation if order matters and a combination if order
does not matter.
• There are 12 different-colored cubes in a bag. How
many ways can Randall draw a set of 4 cubes from the
bag?
• The swim team has 8 swimmers. Two swimmers will be
selected to swim in the first heat. How many ways can
the swimmers be selected?
Exanmple:

5. Definitions
Sample space: A list of all possible outcomes of
an experiment.
Sample point: Each individual outcome in the
sample space.
Tree diagrams are helpful in determining sample
spaces.
12.5
-5

6. Example 1: Selecting Balls without
Replacement
Two balls are to be selected without replacement
from a bag that contains one red, one blue, one
green and one orange ball.
a) Use the counting principle to determine the
number of points in the sample space.
Solution
There are 4 • 3 = 12 sample points.
12.5
-6

7. Example 1: Selecting Balls without
Replacement
b) Construct a tree diagram and list the sample
space.
Solution
The first ball selected can be red, blue, green, or
orange.
Since this experiment is done without
replacement, the same colored ball cannot be
selected twice.
See the tree diagram on the next slide.
12.5
-7

8. 12.5
-8

9. Example 2: Selecting Ticket Winners
A radio station has two tickets to give away to a
Bon Jovi concert. It held a contest and narrowed
the possible recipients down to four people:
Christine (C), Mike Hammer (MH), Mike Levine
(ML), and Phyllis (P).
The names of two of these four people will be
selected at random from a hat, and the two
people selected will be awarded the tickets.
12.5
-9

10. Example 2: Selecting Ticket Winners
a) Use the counting principle to determine the
number of points in the sample space.
Solution
There are 4 • 3 = 12 sample points in the sample
space.
12.5
-10

11. Example 2: Selecting Ticket Winners
b) Construct a tree diagram and list the sample
space.
Solution
See tree diagram on the next slide.
12.5
-11

12. b) Construct a tree diagram and list
the sample space.
12.5
-12

13. Indepe
ndent
red
blue
First Choice
Second Choice
3 3
P(red and red) =
10 10
9
100
x 
7
10
red
blue
red
blue
3
10 3 7
P(red and blue) =
10 10
21
100
x 
7 3
P(blue and red) =
10 10
21
100
x 
7 7
P(blue and blue) =
10 10
49
100
x 
3
10
7
10
3
10
7
10
Peter has ten coloured cubes in a bag. Three of the
cubes are red and 7 are blue. He removes a cube at
random from the bag and notes the colour before
replacing it. He then chooses a second cube at
random. Record the information in a tree diagram.

14. Q1 beads
black
green
First Choice Second Choice
5
9
black
green
black
green
4
9 4 5
P(black and green) =
9 9
20
81
x 
5 5
P(green and green) =
9 9
25
81
x 
4
9
5
9
4
9
5
9
4 4
P(black and black) =
9 9
16
81
x 
5 4
P(green and black) =
9 9
20
81
x 
Rebecca has nine coloured beads in a bag. Four of the beads are
black and the rest are green. She removes a bead at random
from the bag and notes the colour before replacing it. She then
chooses a second bead. Draw a tree diagram showing all possible
outcomes.

15. Q2 Coins
head
tail
First Coin
Second Coin
1
2
head
tail
head
tail
1
2 1 1
P(head and tail)
2
1
=
2 4
x 
1 1
P(tail and tail)
2
1
=
2 4
x 
1
2
1
2
1
2
1
2
1 1
P(head and head)
2
1
=
2 4
x 
1 1
P(tail and head)
2
1
=
2 4
x 
P(2 heads) = ¼ P(head and a tail or a tail and a head) = ½
Peter tosses two coins. (a) Draw a tree diagram to show all
possible outcomes. (b) Use your tree diagram to find the
probability of getting (i) 2 Heads (ii) A head or a tail in
any order.

16. Q3
Sports
Becky
Win
Becky
Win
Peter
Win
Becky
Win
Race
Tennis
Peter
Win
Peter
Win
0.4
0.7
0.6
0.3
0.3
0.7
0.4 x 0.3 = 0.12
0.4 x 0.7 = 0.28
0.6 x 0.3 = 0.18
0.6 x 0.7 = 0.42
P(Win and Win) for Peter = 0.12 P(Lose and Win) for Becky = 0.28
Peter and Becky run a race and play a tennis match. The probability
that Peter wins the race is 0.4. The probability that Becky wins the
tennis is 0.7. (a) Complete the tree diagram below. (b) Use your tree
diagram to calculate (i) the probability that Peter wins both events.
(ii) The probability that Becky loses the race but wins at tennis.
win

17. H
T
H
H
H
H
H
H
T
T
T
T
T
T
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Examples:
A coin is tossed 3 times, show the outcomes using a tree
diagram.
Pr ( 2 heads) =
Pr(HHT)+Pr(HTH)+
Pr(THH) =

18. 2. A coin is tossed and then a die is rolled. Draw the tree diagram
H
T
1
2
3
4
5
6
1
2
3
4
5
6

19. 19
What is the probability of:
tossing a head then rolling a five?
tossing a tail then rolling an even?
tossing a tail then rolling a seven?

20. 3. Consider the following tree diagram:
1
4
3
4
1
4
3
4
2
3
1
3
Red
Red
Red
Blue
Blue
Blue
What is the probability of:
a)Drawing two reds?
b)Drawing a red then a blue?
c)Drawing 2 blues?
d)Drawing no blues?
e)Drawing 1 blue?
6
1
4
1
3
2

2
1
4
3
3
2

4
1
4
3
3
1

6
1
4
1
3
2

12
7
12
1
2
1
4
1
3
1
4
3
3
2
This can be done 2 ways: RB or BR:

21. Dependen
t
red
blue
First Choice
Second Choice
3 2
P(red and red) =
10 9
6
90
x 
7
10
red
blue
red
blue
3
10 3 7
P(red and blue) =
10 9
21
90
x 
7 3
P(blue and red) =
10 9
21
90
x 
7 6
P(blue and blue) =
10 9
42
90
x 
2
9
7
9
3
9
6
9
Peter has ten coloured cubes in a bag. Three of the cubes
are red and seven are blue. He removes a cube at random
from the bag and notes the colour but does not replace it.
He then chooses a second cube at random. Record the
information in a tree diagram.

22. Q4 beads
black
green
First Choice
Second Choice
5
9
black
green
black
green
4
9 4 5
P(black and green) =
9 8
20
72
x 
5 4
P(green and green) =
9 8
20
72
x 
3
8
5
8
4
8
4
8
4 3
P(black and black) =
9 8
12
72
x 
5 4
P(green and black) =
9 8
20
72
x 
Rebecca has nine coloured beads in a bag. Four of the beads are
black and the rest are green. She removes a bead at random from
the bag and does not replace it. She then chooses a second bead.
(a) Draw a tree diagram showing all possible outcome (b) Calculate
the probability that Rebecca chooses: (i) 2 green beads (ii) A black
followed by a green bead.

23. Milk
Dark
First Pick
Second Pick
12
30
Milk
Dark
Milk
Dark
18
30
18 17
P(milk and milk) =
306
87030 29
x 
17
29
12
29
18
29
11
29
18 12
P(milk and dark) =
216
87030 29
x 
12 18
P(dark and milk) =
216
87030 29
x 
12 11
P(dark and dark) =
132
87030 29
x 
Lucy has a box of 30 chocolates. 18 are milk chocolate and
the rest are dark chocolate. She takes a chocolate at random
from the box and eats it. She then chooses a second.
Draw a tree diagram to show all the possible outcomes.

24. 24
Examples:
A family has three children. What is the probability that:
a) They are all boys?
b) The 1st is a boy and the 2nd and the 3rd are girls?
c) There is one boy and two girls?

25. B
B
B
B
B
B
B
G
G
G
G
G
G
G

26. 26
Example
A mathematics student calculates his chances of passing
the next test according to the results on earlier tests. If
he passed the last test he thinks his chances are 0.7
of passing the next test. If he failed the last test he
estimates that the probability of passing the next
test is 0.5. Draw a probability tree diagram to illustrate
the possible results obtained on the next two tests,
given that he failed the previous test. Find the
probability that on the next two tests the student will:
(a)pass both
(b)pass the first but not the second
(c) fail the first and pass the second
(d) fail both

27. Pass
Pass
Pass
Fail
Fail
Fail
Fail
0.7
0.5
0.5
0.5
0.5
0.3

28. 28
a) Pr(PP) = 0.5 x 0.7 = 0.35
b) Pr(PF) = 0.5 x 0.3 = 0.15
c) Pr(FP) = 0.5 x 0.5 = 0.25
d) Pr(FF) = 0.5 x 0.5 = 0.25

29. 29
Example.
From an urn containing 7 blue and 3 red balls, 2 balls are
taken at random
(i) with replacement (Independent)
(ii) without replacement (Dependent)
Find the probability that:
(a) both balls are blue
(b) the first ball is red and the second is blue
(c) one is red and the other is blue

30. 30
(i) (ii)
a)Pr(BB)
b) Pr(RB)
c) Pr(RB or BR)

31. Show the sample space for tossing one
penny and rolling one die.
(H = heads, T = tails)
• Sample Space:
{H1, H2, H3, H4, H5, H6,
T1, T2, T3, T4, T5, T6}
•The probability of each of these "path" outcomes
Is 1/2 • 1/6 = 1/12.

33. There are three children in a family.
How many outcomes can represent the order in
which the children may have been born in relation to
their gender.
How many outcomes will be in the sample space
indicating the gender of the children?
Assume that the probability of male (M) and the
probability of female (F) are each 1/2.

34. Sample Space:
{MMM,
MMF,
MFM,
MFF,
FMM,
FMF,
FFM,
FFF}
• There are 8 outcomes in the
sample space.
• The probability of each outcome
is
1/2 • 1/2 • 1/2 = 1/8.

37. 37

38. Suppose you were going to flip a coin and roll a fair
die.
Draw a tree-diagram of the sample space, list the sample
space. How many outcomes are in the sample space?