Probability 101
CHAPTER THREE
Probability
Objectives
At the end of this chapter, you should be able to:
i. Define probability.
ii. Describe the classical, the relative frequency, and the subjective approaches to
probability.
iii. Calculate probabilities, applying the rules of addition and multiplication.
iv. Determine the number of possible permutations and combinations.
v. Calculate a probability using Baye’s Theorem.
vi. Define a probability distribution.
vii. Distinguish between a discrete probability distribution and a continuous probability
distribution.
viii Calculate the mean, variance and standard deviation of a probability distribution.
ix. Construct binomial and poisson distribution.
TE X T
x. Determine which probability distribution to use in a given situation.
Fast Forward: Probability, or chane, is a way of expressing knowledge or belief of the possibility
or the strength of the possibility that an event will occur.
STU D Y
Introduction
Probability is a measure of likelihood, the possibility or chance that an event will happen in future.
It can be considered as a quantification of uncertainty.
Uncertainty may also be expressed as likelihood, chance or risk theory. It is a branch of
mathematics concerned with the concept and measurement of uncertainty.
Much in life is characterised by uncertainty in actual decision making.
Probability can only assume a value between 0 and 1 inclusive. The closer a probability is to zero
the more improbable that something will happen. The closer the probability is to one the more
likely it will happen.
Definitions of key terms
Random experiment results in one of a number of possible outcome e.g. tossing a coin
Outcome is the result of an experiment e.g. head up, gain, loss, etc. Specific outcomes are
known as events.
, 102 q u a n t i tat i v e t e c h n i q u e s
Trial- Each repetition of an experiment can be thought of as a trial which has an observable
outcome e.g. in tossing a coin, a single toss is a trial which has an outcome as either head or
tail
Sample space is the set of all possible outcomes in an experiment e.g. a single toss of a coin,
S=(H,T). The sample space can be finite or infinite. A finite sample space has a finite number of
possible outcomes e.g. in tossing a coin only 2 outcomes are possible.
An infinite sample space has an infinite number of possible outcomes e.g. time between arrival
of telephone calls and telephone exchange.
An Event of an experiment is a subset of a sample space e.g in tossing a coin twice S= (HH, HT,
TH, TT) HH is a subset of a sample space.
Mutually exclusive event - A set of events is said to be mutually exclusive if the occurrence of
any one of the events precludes the occurrence of other events i.e the occurrence of any one
event means none of the others can occur at the same time e.g. the events head and tail are
mutually exclusive
Collectively exclusive event - A set of events is said to be collectively exclusive if their union
accounts for all possible outcomes i.e. one of their events must occur when an experiment is
conducted.
Favourable events refers to the number of possible occurrences of a given event in an experiment
e.g. if we pick a card from a deck of 52 cards the number favorable to a red card is 26, in tossing
TE X T
a coin the number favourable to a head is one.
Independent events – Events are independent if the happening or non-happening of one has no
effect on the future happening of another event. E.g. in tossing two times of a coin, the outcome
STU D Y
of 1st toss does not affect 2nd toss.
Equally likely events – Events are equally likely if the happening of one is not favoured over the
happening of others. In tossing a coin the tail and head are equally likely.
OTHER CONCEPTS
Unconditional and conditional probabilities – with unconditional probability we express
probability of an event as a ratio of favourable outcomes to the number of all possible
outcomes.
A conditional probability is the probability that a second event occurs if the first event has
already occurred.
Joint probability – joint probability gives the probability of the joint or simultaneous occurrence
of two or more characteristics.
Marginal probability – is the sum of two or more joint probabilities taken over all values of one
or more variables. It is the probability that the results when we ignore one or more criteria of
classification when computing probability.
, Probability 103
Industry context
Probability is used throughout business to evaluate financial risks and decision-making.
Every decision made by management carries some chance for failure, so probabiity
analysis is conducted formally.
In many natural processes, random variation conforms to a particular probability
distribution known as the normal distribution, which is the most commonly observed
probability distribution.
EXAM CONTEXT
The probability topic has been examined previously:
6/06, 12/05, 6/05, 12/04, 6/04, 6/03, 12/02, 6/02, 12/01, 12/00, 6/00
Laws of Probability
1. Rules of Addition
TE X T
a) Special rule of addition
If two events A and B are mutually exclusive the probability of one or other occurring is equal to
STU D Y
the sum of their probability
P (A or B) = P (A) + P (B)
P (A or B or C) = P (A) + P (B) + P(C)
Illustration
An automatic plastic bag - a mixture of beans, broccoli and other vegetables, most of the
bags contain the correct weight but because of slight variations in the size of beans and other
vegetables. A package may be slightly under or overweight. A check of 4,000 packages of past
reveals the following:
Weight Event No of packages
Underweight A 100
Satisfactory B 3600
Overweight C 300
What is the probability that a particular package will be either underweight or overweight?
The two events are mutually exclusive
P (A or C) = P (A) + P (C)
P(A) =100/4000
P(C) =300/4000
P(A or C) =400/4000 = 1/10 =0.1
CHAPTER THREE
Probability
Objectives
At the end of this chapter, you should be able to:
i. Define probability.
ii. Describe the classical, the relative frequency, and the subjective approaches to
probability.
iii. Calculate probabilities, applying the rules of addition and multiplication.
iv. Determine the number of possible permutations and combinations.
v. Calculate a probability using Baye’s Theorem.
vi. Define a probability distribution.
vii. Distinguish between a discrete probability distribution and a continuous probability
distribution.
viii Calculate the mean, variance and standard deviation of a probability distribution.
ix. Construct binomial and poisson distribution.
TE X T
x. Determine which probability distribution to use in a given situation.
Fast Forward: Probability, or chane, is a way of expressing knowledge or belief of the possibility
or the strength of the possibility that an event will occur.
STU D Y
Introduction
Probability is a measure of likelihood, the possibility or chance that an event will happen in future.
It can be considered as a quantification of uncertainty.
Uncertainty may also be expressed as likelihood, chance or risk theory. It is a branch of
mathematics concerned with the concept and measurement of uncertainty.
Much in life is characterised by uncertainty in actual decision making.
Probability can only assume a value between 0 and 1 inclusive. The closer a probability is to zero
the more improbable that something will happen. The closer the probability is to one the more
likely it will happen.
Definitions of key terms
Random experiment results in one of a number of possible outcome e.g. tossing a coin
Outcome is the result of an experiment e.g. head up, gain, loss, etc. Specific outcomes are
known as events.
, 102 q u a n t i tat i v e t e c h n i q u e s
Trial- Each repetition of an experiment can be thought of as a trial which has an observable
outcome e.g. in tossing a coin, a single toss is a trial which has an outcome as either head or
tail
Sample space is the set of all possible outcomes in an experiment e.g. a single toss of a coin,
S=(H,T). The sample space can be finite or infinite. A finite sample space has a finite number of
possible outcomes e.g. in tossing a coin only 2 outcomes are possible.
An infinite sample space has an infinite number of possible outcomes e.g. time between arrival
of telephone calls and telephone exchange.
An Event of an experiment is a subset of a sample space e.g in tossing a coin twice S= (HH, HT,
TH, TT) HH is a subset of a sample space.
Mutually exclusive event - A set of events is said to be mutually exclusive if the occurrence of
any one of the events precludes the occurrence of other events i.e the occurrence of any one
event means none of the others can occur at the same time e.g. the events head and tail are
mutually exclusive
Collectively exclusive event - A set of events is said to be collectively exclusive if their union
accounts for all possible outcomes i.e. one of their events must occur when an experiment is
conducted.
Favourable events refers to the number of possible occurrences of a given event in an experiment
e.g. if we pick a card from a deck of 52 cards the number favorable to a red card is 26, in tossing
TE X T
a coin the number favourable to a head is one.
Independent events – Events are independent if the happening or non-happening of one has no
effect on the future happening of another event. E.g. in tossing two times of a coin, the outcome
STU D Y
of 1st toss does not affect 2nd toss.
Equally likely events – Events are equally likely if the happening of one is not favoured over the
happening of others. In tossing a coin the tail and head are equally likely.
OTHER CONCEPTS
Unconditional and conditional probabilities – with unconditional probability we express
probability of an event as a ratio of favourable outcomes to the number of all possible
outcomes.
A conditional probability is the probability that a second event occurs if the first event has
already occurred.
Joint probability – joint probability gives the probability of the joint or simultaneous occurrence
of two or more characteristics.
Marginal probability – is the sum of two or more joint probabilities taken over all values of one
or more variables. It is the probability that the results when we ignore one or more criteria of
classification when computing probability.
, Probability 103
Industry context
Probability is used throughout business to evaluate financial risks and decision-making.
Every decision made by management carries some chance for failure, so probabiity
analysis is conducted formally.
In many natural processes, random variation conforms to a particular probability
distribution known as the normal distribution, which is the most commonly observed
probability distribution.
EXAM CONTEXT
The probability topic has been examined previously:
6/06, 12/05, 6/05, 12/04, 6/04, 6/03, 12/02, 6/02, 12/01, 12/00, 6/00
Laws of Probability
1. Rules of Addition
TE X T
a) Special rule of addition
If two events A and B are mutually exclusive the probability of one or other occurring is equal to
STU D Y
the sum of their probability
P (A or B) = P (A) + P (B)
P (A or B or C) = P (A) + P (B) + P(C)
Illustration
An automatic plastic bag - a mixture of beans, broccoli and other vegetables, most of the
bags contain the correct weight but because of slight variations in the size of beans and other
vegetables. A package may be slightly under or overweight. A check of 4,000 packages of past
reveals the following:
Weight Event No of packages
Underweight A 100
Satisfactory B 3600
Overweight C 300
What is the probability that a particular package will be either underweight or overweight?
The two events are mutually exclusive
P (A or C) = P (A) + P (C)
P(A) =100/4000
P(C) =300/4000
P(A or C) =400/4000 = 1/10 =0.1
