PROBABILITY
Definition. It’s a value between zero and one inclusive, describing the relative possibility
(chance or likelihood) that an event will occur.
Terms used in probability
Experiment: It’s a process or course of action that results in one of a number of possible
outcomes. The outcome that occurs cannot be predicted with certainty e.g. tossing a coin.
Outcome: It’s a particular result of an experiment e.g. getting a head
Event: it’s a collection of one or more outcomes of an experiment.
Sample space: It’s a list of all possible outcomes of the experiment. The outcomes listed
must be mutually exclusive and exhaustive.
Mutually exclusive events: They are events which cannot occur at the same time i.e. when
one event occurs, none of the other events can occur at the same time.
Independent events: -two events are independent if the occurrence of one event does not
alter the probability of the other event.
Collectively exhaustive: At least one of the events must occur when an experiment is
conducted.
Approaches to Assigning Probability
There are three approaches to probability
(a) Classical probability: It’s based on the assumption that the outcomes of an experiment are
equally likely.
Probability of an event =
(b) Empirical probability: the probability of an event occurring is determined by observing
what fraction of the time similar events happened in the past. Probability is based on relative
frequencies.
Probability of an event =
(c) Subjective probability: The likelihood of a particular event happening is assigned by an
individual based on whatever information is available.
Rules for Computing Probabilities
1. Rule of addition
If two events A and B are mutually exclusive, the probability of one or the other event’s
occurring equals the sum of their probabilities.
If the two events are not mutually exclusive
Example
In a class of 20 children, 4 of the 9 boys and 3 of the 11 girls are in the athletics team. A person
from the class is chosen to be in the ‘egg and spoon’ race on the sports day. Find the probability
that the person chosen is:-
(a) In the athletics team
(b) Female
(c) A female member of the athletics team
, (d) A female or in the athletics team
2. The complement Rule
It is used to determine the probability of an event occurring by subtracting the probability of the
event not occurring from one.
Examples
1. In a survey, 15% of the participants said that they had never bought lottery tickets or
premium bonds, 73% had bought lottery tickets and 49% had bought premium bonds. Find
the probability that a person chosen at random from those taking part in the survey had
bought:-
i. Lottery tickets or premium bonds
ii. Lottery tickets and premium bonds
iii. Lottery tickets only
2. A selected group of employees of Worldwide enterprises is to be surveyed about a new pension
plan. In-depth interviews are to be conducted with each employee selected in the sample. The
employees are classified as follows:
Classification Number of employees
Supervisors 120
Maintainance 50
Production 1460
Management 302
Secretarial 68
What is the probability that the first person selected is:
i. Either in maintainance or a secretary
ii. Not in management
3. The probability that a contractor will get a plumbing contract is 2 3 and the probability that
he will not get an electric contract is 5 9 . If the probability of getting at least one contract is
4 , what is the probability that he will get both?
5
3. Rules of multiplication
It is used to find the joint occurrence of two or more events.
If two events A and B are independent
If two events A and B are dependent
P( B / A) is the probability of B occurring given that A has already occurred
Conditional probability
It is the probability of a particular event occurring given that another event has occurred.
Example
Definition. It’s a value between zero and one inclusive, describing the relative possibility
(chance or likelihood) that an event will occur.
Terms used in probability
Experiment: It’s a process or course of action that results in one of a number of possible
outcomes. The outcome that occurs cannot be predicted with certainty e.g. tossing a coin.
Outcome: It’s a particular result of an experiment e.g. getting a head
Event: it’s a collection of one or more outcomes of an experiment.
Sample space: It’s a list of all possible outcomes of the experiment. The outcomes listed
must be mutually exclusive and exhaustive.
Mutually exclusive events: They are events which cannot occur at the same time i.e. when
one event occurs, none of the other events can occur at the same time.
Independent events: -two events are independent if the occurrence of one event does not
alter the probability of the other event.
Collectively exhaustive: At least one of the events must occur when an experiment is
conducted.
Approaches to Assigning Probability
There are three approaches to probability
(a) Classical probability: It’s based on the assumption that the outcomes of an experiment are
equally likely.
Probability of an event =
(b) Empirical probability: the probability of an event occurring is determined by observing
what fraction of the time similar events happened in the past. Probability is based on relative
frequencies.
Probability of an event =
(c) Subjective probability: The likelihood of a particular event happening is assigned by an
individual based on whatever information is available.
Rules for Computing Probabilities
1. Rule of addition
If two events A and B are mutually exclusive, the probability of one or the other event’s
occurring equals the sum of their probabilities.
If the two events are not mutually exclusive
Example
In a class of 20 children, 4 of the 9 boys and 3 of the 11 girls are in the athletics team. A person
from the class is chosen to be in the ‘egg and spoon’ race on the sports day. Find the probability
that the person chosen is:-
(a) In the athletics team
(b) Female
(c) A female member of the athletics team
, (d) A female or in the athletics team
2. The complement Rule
It is used to determine the probability of an event occurring by subtracting the probability of the
event not occurring from one.
Examples
1. In a survey, 15% of the participants said that they had never bought lottery tickets or
premium bonds, 73% had bought lottery tickets and 49% had bought premium bonds. Find
the probability that a person chosen at random from those taking part in the survey had
bought:-
i. Lottery tickets or premium bonds
ii. Lottery tickets and premium bonds
iii. Lottery tickets only
2. A selected group of employees of Worldwide enterprises is to be surveyed about a new pension
plan. In-depth interviews are to be conducted with each employee selected in the sample. The
employees are classified as follows:
Classification Number of employees
Supervisors 120
Maintainance 50
Production 1460
Management 302
Secretarial 68
What is the probability that the first person selected is:
i. Either in maintainance or a secretary
ii. Not in management
3. The probability that a contractor will get a plumbing contract is 2 3 and the probability that
he will not get an electric contract is 5 9 . If the probability of getting at least one contract is
4 , what is the probability that he will get both?
5
3. Rules of multiplication
It is used to find the joint occurrence of two or more events.
If two events A and B are independent
If two events A and B are dependent
P( B / A) is the probability of B occurring given that A has already occurred
Conditional probability
It is the probability of a particular event occurring given that another event has occurred.
Example
