Probability is a quantitative measure of the likelihood of
occurrence for chance events. The concept of pure chance is not absolutely true because all events are predetermined. Humans
use chance or probability estimates because of their limited knowledge. What appears random or chance to humans has an underlying
deterministic order known only to the Creator. The consistency of probabilities and predictions is based on underlying physical
laws that govern the universe.
Probability is commonly defined
as relative frequency of an event on repeated trials under the same conditions. Each
possible outcome is called a sample point. The set of all possible outcomes is called the probability space, S. If the probability
space consists of a finite number of equally likely events, probability of event ‘A’ is defined as: Pr (A) = n
(A) / n (S) where n(A) = number of events of type A and n(S) = the total number in the probability space. Special mathematical
techniques called arrangements, permutations and combinations, can enable us calculate the probability space theoretically
without having to carry out the trials.
2.0 CLASSIFICATION OF PROBABILITY
Probability can be subjective (based personal feelings or
intuition) or objective (based on real data or experience). Objective probability can be measured or computed. Prior probability is knowable or calculable without experimentation. Posterior probability
is calculable from results of experimentation. Bayesian probability combines
prior probability (objective, subjective, or a belief) with new data (from experimentation) to reach a conclusion called posterior
probability. Bayesian probability is a good representation of how conclusions are made from empirical observation in real
life. Conditional probability is employed when there is partial information or when
we want to make probability computations easier by assuming conditionality. In conditional probability, the event depends
on occurrence of a previous event.
3.0 TYPES OF EVENTS
On the scale of exclusion, events are classified as mutually
exclusive or nonmutually exclusive. Mutually exclusive events are those that cannot occur together like being dead and being
alive. Not all mutually exclusive events are equally likely. On the scale of independence, events are classified as independent
or dependent. Under independence, the occurrence of one event is not affected by occurrence or nonoccurrence of another.
Independent events can occur at the same instant or subsequently. Some independent events are equally likely while others
are not. On the scale of exhaustion, two events A and B are said to be exhaustive if between them they occupy all the probability
space ie A U B = S and Pr (A U B) = 1.
Confusion occurs between mutually exclusive and independent
events. Mutually exclusive events cannot both occur at the same time i.e. Pr (A n B) = 0. Mutually exclusive events cannot
be independent of one another because the occurrence of one will prevent the other one from occurring. Independent events
can both occur at the same time but the occurrence of one is not affected by the occurrence of the other i.e. Pr (A n B) =
Pr (A) x Pr (B).
4.0 LAWS OF PROBABILITY and MATHEMATICAL PROPERTIES
The total probability space is equal to 1.0. This is stated
mathematically as Pr (S) = 1.0. If the probability of occurrence is p, the probability of nonoccurrence is 1p. If the sample
space has equally likely outcomes, then Pr (A) = n (A) / n(S). For two events p and q, p + q =1 where p is probability of
occurrence of an event and q is probability of its nonoccurrence. Note that certainty has a probability of 1. This can be
restated as Pr (Ā) = 1 – Pr (A) or as Pr (A) + Pr (Ā) = 1. The additive law, also called the ‘OR’
rule, refers to occurrence of any or both events and is stated as Pr (A u B) = Pr (A) + Pr (B)  Pr (A n B) where Pr (A n
B) = 0 for mutually exclusive events. The multiplicative law for independent events refers to the joint occurrence of the
events or the ‘AND’ rule and is stated as Pr (A n B) = Pr (A) x Pr (B). The range of probability is 0.0 to 1.0
and cannot be negative. Pr = 0.0 means the event is impossible. Pr = 1.0 means the event is absolutely certain. The odds of
an event can be defined as {Pr (A)} / {1 – Pr (A)}.
5.0 USES OF PROBABILITY
Probability is used in classical statistical inference, Bayesian statistical inference, clinical decisionmaking,
queuing theories, and probability trees.
Computation exercises
 If the probability of success in a cardiac
operation is 40%, the probability of success in 5 consecutive such operations on 5 different patients is (a) 0.4 + 0.4 + 0.4
+ 0.4 + 0.4 = 2.0 (b) 0.4 x 5 (c) 0.4 x 0.4 x 0.4 x 0.4 x 0.4
 Explain how you can compute the probability
of measles in a child with both fever and a rash and the Pr (fever) and Pr (rash) are known from the database.
 If 2 independent events have probability
1/4 each, the probability of their joint occurrence is (a) 1/4 x 1/4 (b) 1/4 + 1/4
 Compute the probability that in a family
of 4, the 4^{th} child is a boy
 In a medical class of 15, 8 are male
and 10 are from the BruneiMuara. Compute the probability that a student selected at random will be (a) a male (b) from BruneiMuara
(c) is both a male and from the east coast
Practical assignments
 Toss 1 coin 20 times and compute Pr (heads)
and Pr (tails). Draw a graph of Pr (heads) vs. # tosses.
2.
Toss 2 coins at a time. The successful event is when
both show heads. Complete the following table showing the probability of success for different throws. Complete the table
and also draw a graph.
# Trials 
#Total outcomes 
#Successes 
Probability 
1 



2 



4 



4 



5 



Key words and terms: ComputerAssisted Diagnosis, Decision Support Techniques;
Event, complimentary; Event, independent; Event, mutually exclusive; Gambler's fallacy; Law, ‘and law; Law, ‘or’
law; Law, addition law; Law, multiplicative law; Probability space; Probability trees; a priori probability; Anterior probability;
Bayesian probability; Probability, classical probability; conditional probability; empirical probability; frequentist probability;
laws of probability; objective probability; posterior probability; subjective probability; theoretical probability