Sample Spaces

2
CHAPTER
Sample Spaces
Introduction
In order to compute classical probabilities, you need to find the sample
space for a probability experiment. In the previous chapter, sample spaces
were found by using common sense. In this chapter two specific devices
will be used to find sample spaces for probability experiments. They are tree
diagrams and tables.
Tree Diagrams
A tree diagram consists of branches corresponding to the outcomes of two or
more probability experiments that are done in sequence.
In order to construct a tree diagram, use branches corresponding to the
outcomes of the first experiment. These branches will emanate from a single
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point. Then from each branch of the first experiment draw branches that
represent the outcomes of the second experiment. You can continue the
process for further experiments of the sequence if necessary.
EXAMPLE: A coin is tossed and a die is rolled. Draw a tree diagram and
find the sample space.
SOLUTION:
1. Since there are two outcomes (heads and tails for the coin), draw two
branches from a single point and label one H for head and the other
one T for tail.
2. From each one of these outcomes, draw and label six branches repre-
senting the outcomes 1, 2, 3, 4, 5, and 6 for the die.
3. Trace through each branch to find the outcomes of the experiment.
See Figure 2-1.
Hence there are twelve outcomes. They are H1, H2, H3, H4, H5, H6, T1,
T2, T3, T4, T5, and T6.

Once the sample space has been found, probabilities for events can be
computed.
Fig. 2-1.
CHAPTER 2 Sample Spaces
23
EXAMPLE: A coin is tossed and a die is rolled. Find the probability of
getting
a. A head on the coin and a 3 on the die.
b. A head on the coin.
c. A 4 on the die.
SOLUTION:
a. Since there are 12 outcomes in the sample space and only one way to
get a head on the coin and a three on the die,
PðH3Þ¼
1
12
b. Since there are six ways to get a head on the coin, namely H1, H2, H3,
H4, H5, and H6,
Pðhead on the coin) ¼
6
12
¼
1
2
c. Since there are two ways to get a 4 on the die, namely H4 and T4,
Pð4 on the die) ¼
2
12
¼
1

6
EXAMPLE: Three coins are tossed. Draw a tree diagram and find the sample
space.
SOLUTION:
Each coin can land either heads up (H) or tails up (T); therefore, the tree
diagram will consist of three parts and each part will have two branches.
See Figure 2-2.
Fig. 2-2.
CHAPTER 2 Sample Spaces
24
Hence the sample space is HHH, HHT, HTH, HTT, THH, THT, TTH,
TTT.
Once the sample space is found, probabilities can be computed.
EXAMPLE: Three coins are tossed. Find the probability of getting
a. Two heads and a tail in any order.
b. Three heads.
c. No heads.
d. At least two tails.
e. At most two tails.
SOLUTION:
a. There are eight outcomes in the sample space, and there are three ways
to get two heads and a tail in any order. They are HHT, HTH,
and THH; hence,
P(2 heads and a tail) ¼
3
8
b. Three heads can occur in only one way; hence
PðHHHÞ¼
1
8

c. The event of getting no heads can occur in only one way—namely
TTT; hence,
PðTTTÞ¼
1
8
d. The event of at least two tails means two tails and one head or three
tails. There are four outcomes in this event—namely TTH, THT,
HTT, and TTT; hence,
P(at least two tails) ¼
4
8
¼
1
2
e. The event of getting at most two tails means zero tails, one tail,
or two tails. There are seven outcomes in this event—HHH, THH,
HTH, HHT, TTH, THT, and HTT; hence,
P(at most two tails) ¼
7
8
When selecting more than one object from a group of objects, it is
important to know whether or not the object selected is replaced before
drawing the second object. Consider the next two examples.
CHAPTER 2 Sample Spaces
25
EXAMPLE: A box contains a red ball (R), a blue ball (B), and a yellow ball
(Y). Two balls are selected at random in succession. Draw a tree diagram and
find the sample space if the first ball is replaced before the second ball is
selected.
SOLUTION:

There are three ways to select the first ball. They are a red ball, a blue ball, or
a yellow ball. Since the first ball is replaced before the second one is selected,
there are three ways to select the second ball. They are a red ball, a blue ball,
or a yellow ball. The tree diagram is shown in Figure 2-3.
The sample space consists of nine outcomes. They are RR, RB, RY, BR,
BB, BY, YR, YB, YY. Each outcome has a probability of
1
9
:
Now what happens if the first ball is not replaced before the second ball
is selected?
EXAMPLE: A box contains a red ball (R), a blue ball (B), and a yellow ball
(Y). Two balls are selected at random in succession. Draw a tree diagram and
find the sample space if the first ball is not replaced before the second ball is
selected.
SOLUTION:
There are three outcomes for the first ball. They are a red ball, a blue ball, or
a yellow ball. Since the first ball is not replaced before the second ball is
drawn, there are only two outcomes for the second ball, and these outcomes
depend on the color of the first ball selected. If the first ball selected is blue,
then the second ball can be either red or yellow, etc. The tree diagram is
shown in Figure 2-4.
Fig. 2-3.
CHAPTER 2 Sample Spaces
26
The sample space consists of six outcomes, which are RB, RY, BR, BY,
YR, YB. Each outcome has a probability of
1
6
:

PRACTICE
1. If the possible blood types are A, B, AB, and O, and each type can be
Rh
þ
or Rh
À
, draw a tree diagram and find all possible blood types.
2. Students are classified as male (M) or female (F), freshman (Fr),
sophomore (So), junior (Jr), or senior (Sr), and full-time (Ft) or part-
time (Pt). Draw a tree diagram and find all possible classifications.
3. A box contains a $1 bill, a $5 bill, and a $10 bill. Two bills are selected
in succession with replacement. Draw a tree diagram and find the
sample space. Find the probability that the total amount of money
selected is
a. $6.
b. Greater than $10.
c. Less than $15.
4. Draw a tree diagram and find the sample space for the genders of the
children in a family consisting of 3 children. Assume the genders are
equiprobable. Find the probability of
a. Three girls.
b. Two boys and a girl in any order.
c. At least two boys.
5. A box contains a white marble (W), a blue marble (B), and a green
marble (G). Two marbles are selected without replacement. Draw a
tree diagram and find the sample space. Find the probability that one
marble is white.
Fig. 2-4.
CHAPTER 2 Sample Spaces
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ANSWERS
1.
2.
Fig. 2-6.
Fig. 2-5.
CHAPTER 2 Sample Spaces
28
3.
There are nine outcomes in the sample space.
a. Pð$6Þ¼
2
9
since $1+$5, and $5+$1 equal $6.
b. Pðgreater than $10) ¼
5
9
since there are five ways to get a sum
greater than $10.
c. Pðless than $15) ¼
6
9
¼
2
3
since there are six ways to get a sum lesser
than $15.
4.
There are eight outcomes in the sample space.
a. Pð3 girls) ¼
1

8
since three girls is GGG.
b. P(2 boys and one girl in any order) ¼
3
8
since there are three ways
to get two boys and one girl in any order. They are BBG, BGB,
and GBB.
Fig. 2-8.
Fig. 2-7.
CHAPTER 2 Sample Spaces
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