Introduction to Probability and
Probability Models
Probability of an Event
A measure of the likelihood that an event will
occur.
Number of Desired Outcomes
P  event  
Total Possibe Outcomes
Example: What is the probability of selecting a heart
from a standard deck of cards?
Number of Hearts
13 1

  0.25  25%
P  heart  
Total Number of Cards 52 4
Independent Events
Two events, A and B, are independent if the fact that A
occurs does not affect the probability of B occurring.
When two events, A and B, are independent, the
probability of both occurring is:
“AND”
=
P  A and B   P  A  P  B 
Multiplication
Rule
Ex: What is the probability of selecting an ace from a
standard deck and rolling a 3 on a standard 6-sided
Selecting a card does not affect rolling a die.
die?
P  Ace   P  3 
These events are independent.
4
52
 
1
6
4
312

1
78
Mutually Exclusive Events
Two events, A and B, are mutually exclusive if they can
not occur at the same time. In other words:
P  A and B   0
When two events, A and B, are mutually exclusive, the
probability of either occurring is:
“OR”
=
P  A or B   P  A  P  B 
Division Rule
Ex: If you select one card from a standard deck, what is
the probability of selecting an ace or selecting the king
of hearts? Notice that selecting an ace AND the king of hearts is impossible
if you select one card. These events are mutually exclusive.

52
P  Ace   P  King of Hearts   4
1
52

5
52
The Complement of an Event
The complement of an event A, typically written Ā, is the
set of all outcomes that are not A.
The probability of an event and its complement always
add up to 1:
 
P  A  P A  1
Ex: When tossing a standard 6-sided die, what is the
probability of not getting a 5?
The event of getting a 5 and the event of not getting a 5 are
complements. The sum of their probabilities is 1.
P  not 5  1  P  5 
1 
1
6
5
6
Example
Your teacher challenges you to a spinner game. You
spin the two spinners with the probabilities listed
below. The first letter should come from Spinner
#1 and the second letter from Spinner #2. Find all
of the possibilities and the probabilities of each
possibility.
One way of
finding an
answer is
listing the
outcomes.
U
I
T
A
F
PI  
1
2
P U  
1
6
P  A   13
P T  
1
4
PF  
3
4
Example: Listing the Outcomes
OUTCOME
Spin an”I”,
then a “T”
List all of
the
possibilities
of the two
spins
IT
IF
UT
UF
AT
AF
PROBABILITY
P(I)xP(T) =
1
2
 14 
1
8
1
2
 34 
3
8
1
6
 14 
1
24
1
6
1
3
 34  18
 14  121
1
3
 34 
Another way of finding an answer is
to use an Area Diagram.
1
4
The spins are
independent.
So we can
multiply the
probabilities.
From the last
slide,
remember:
P(I)=1/2
P(U)=1/6
P(A)=1/3
P(T)=1/4
P(F)=3/4
Example: Area Diagram
Reading the Diagram, the
probability of rolling a “U”
and a “T” is:
P UT  
Spinner #2
Another
T of 1
way
4
finding an
answer is
to use a 3
F
Tree
4
Diagram.
1
24
Spinner #1
I
U
A
1
1
1
2
6
3
IT
UT
AT
1
1
24
1
12
IF
UF
AF
3
1
1
8
8
8
4
Example: Tree Diagram
1
S
T
A
R
T
1
I
3
4
2
1
1
4
1
6
U
3
1
3
Reading the Diagram, the
probability of rolling a “U”
and a “T” is:
P UT   241
4
A
4
4
3
4
T
1 1 1
 
2 4 8
F
T
1 3 3
 
2 4 8
1 1 1
 
6 4 24
F
T
1 3 3 1
 

6 4 24 8
1 1 1
 
3 4 12
F
1 3 3 1
  
3 4 12 4