1. Definitions
Probability is the mathematics of chance.
It tells us the relative frequency with which we can
expect an event to occur
The greater the probability the more
likely the event will occur.
It can be written as a fraction, decimal, percent,
or ratio.
3. Definitions
A probability experiment is an action through which
specific results (counts, measurements, or responses)
are obtained.
The result of a single trial in a probability experiment
is an outcome.
The set of all possible outcomes of a probability
experiment is the sample space, denoted as S.
e.g. All 6 faces of a die: S = { 1 , 2 , 3 , 4 , 5 , 6 }
4. Definitions
Other Examples of Sample Spaces may include:
Lists
Tables
Grids
Venn Diagrams
Tree Diagrams
May use a combination of these
Where appropriate always display your sample space
5. Definitions
An event consists of one or more outcomes and is a
subset of the sample space.
Events are often represented by uppercase letters,
such as A, B, or C.
Notation: The probability that event E will occur is
written P(E) and is read
“the probability of event E.”
6. Definitions
• The Probability of an Event, E:
Consider a pair of Dice
• Each of the Outcomes in the Sample Space are random
and equally likely to occur.
e.g. P( ) =
(There are 2 ways to get one 6 and the other 4)
P(E) =
Number of Event Outcomes
Total Number of Possible Outcomes in S
18
1
36
2
7. Definitions
There are three types of probability
1. Theoretical Probability
Theoretical probability is used when each outcome
in a sample space is equally likely to occur.
P(E) =
Number of Event Outcomes
Total Number of Possible Outcomes in S
The Ultimate probability formula
8. Definitions
There are three types of probability
2. Experimental Probability
Experimental probability is based upon observations
obtained from probability experiments.
The experimental probability of an event
E is the relative frequency of event E
P(E) =
Number of Event Occurrences
Total Number of Observations
9. Definitions
There are three types of probability
3. Subjective Probability
Subjective probability is a probability measure
resulting from intuition, educated guesses, and
estimates.
Therefore, there is no formula to calculate it.
Usually found by consulting an expert.
11. Definitions
Law of Large Numbers.
As an experiment is repeated over and over, the
experimental probability of an event approaches
the theoretical probability of the event.
The greater the number of trials the more likely
the experimental probability of an event will equal
its theoretical probability.
12. Types of Event
• Equally likely
• Not equally likely
• Mutually exclusive
• Not mutually exclusive
• Exhaustive
• Completely exhaustive
• Complementary
13. Mutually Exclusive
Two events A and B are mutually exclusive if and
only if:
In a Venn diagram this means that event A is
disjoint from event B.
A and B are M.E.
0
)
(
B
A
P
A B
14. Not Mutually Exclusive
Two events A and B are non mutually exclusive if
and only if:
In a Venn diagram this means that event A is
not disjoint from event B.
0
)
(
B
A
P
A B
A and B are not M.E.
15. Exhaustive
Two events A and B are exhaustive if and only if:
In a Venn diagram this means that Addition of
event A and event B is equal to entire sample
space “s” .
1
)
(
)
(
S
P
B
A
P
A B
A and B are exhaustive.
16. Completely Exhaustive
A single events A is completely exhaustive if and
only if:
In a Venn diagram this means that event A is
equal to entire sample space “s” .
1
)
(
)
(
S
P
A
P
A is completely exhaustive.
1, 2, 3, 4, 5, 6,
7, 8, 9, 10
A
17. Complimentary Events
The complement of event E is the set of all
outcomes in a sample space that are not included in
event E.
The complement of event E is denoted by
Properties of Probability:
E
or
E
)
(
1
)
(
)
(
1
)
(
1
)
(
)
(
1
)
(
0
E
P
E
P
E
P
E
P
E
P
E
P
E
P
18. Rules of Probability
• Addition rule
• Multiplication
• Independent events
• Conditional rule
• Law of total probability
• Bayes theorem
21. The Addition Rule
The probability that at least one of the events A
or B will occur, P(A or B), is given by:
If events A and B are mutually exclusive, then the
addition rule is simplified to:
This simplified rule can be extended to any number
of mutually exclusive events.
)
(
)
(
)
(
)
(
)
( B
A
P
B
P
A
P
B
A
P
B
or
A
P
)
(
)
(
)
(
)
( B
P
A
P
B
A
P
B
or
A
P
22. Examples
if one card is selected at random from deck of 52 cards,
what is probability that card is club or face card or both?
A pair of dice is thrown find the probability of getting a
total of either 5 or 11?
A class contains 10 men and 20 women half of men and half
of women have brown eyes. Find the probability that person
chosen at random is man or has brown eyes?
In a group of 20 adults , 4 out of the 7 women and 2 out of
the 13 men wear glasses, what is the probability that person
chosen at random from group is a woman or some one who
wear glasses?
23. Examples
• If the probabilities are, respectively, 0.09, 0.15, 0.21, and
0.23 that a person purchasing a new automobile will choose
the color green, white, red, or blue, what is the probability
that a given buyer will purchase a new automobile that comes
in one of those colors?
• Interest centers around the life of an electronic component.
Suppose it is known that the probability that the component
survives for more than 6000 hours is 0.42. Suppose also
that the probability that the component survives no longer
than 4000 hours is 0.04.
(a) What is the probability that the life of the component is
less than or equal to 6000 hours?
(b) What is the probability that the life is greater than
4000 hours?
24. The Multiplication Rule for
Mutually exclusive
If events A and B are mutually exclusive, then the
probability of two events, A and B occurring in a
sequence (or simultaneously) is:
This rule can extend to any number of independent
events.
)
(
)
(
)
(
)
( B
P
A
P
B
A
P
B
and
A
P
Two events are independent if the occurrence of the
first event does not affect the probability of the
occurrence of the second event. More on this later
25. The Multiplication Rule for
non-Mutually exclusive
If events A and B are non-mutually exclusive, then
the probability of two events, A and B is:
This rule can extend to any number of dependent
events.
)
|
(
)
(
)
(
)
( A
B
P
A
P
B
A
P
B
and
A
P
Two events are dependent if the occurrence of the
first event is affect the probability of the
occurrence of the second event.
26. Examples
• Suppose that in a senior college class of 500 students it is
found that 210 smoke, 258 drink alcoholic beverages, 216
eat between meals, 122 smoke and drink alcoholic beverages,
83 eat between meals and drink alcoholic beverages, 97
smoke and eat between meals, and 52 engage in all three of
these bad health practices. If a member of this senior class
is selected at random,
• find the probability that the student
(a) smokes but does not drink alcoholic beverages;
(b) eats between meals and drinks alcoholic beverages but
does not smoke;
(c) neither smokes nor eats between meals.
27. • In a high school graduating class of 100 students, 54 studied
mathematics, 69 studied history, and 35 studied both
mathematics and history. If one of these students is selected at
random, find the probability that
(a) the student took mathematics or history;
(b) the student did not take either of these subjects;
(c) the student took history but not mathematics.
• A small town has one fire engine and one ambulance available for
emergencies. The probability that the fire engine is available
when needed is 0.98, and the probability that the ambulance is
available when called is 0.92. In the event of an injury resulting
from a burning building, find the probability that both the
ambulance and the fire engine will be available, assuming they
operate independently.
