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Quantitative Techniques – I

Notes Often, in our day-to-day life, we hear sentences like ‘it may rain today’, ‘Mr X has fifty-fifty
chances of passing the examination’, ‘India may win the forthcoming cricket match against Sri
lanka’, ‘the chances of making profits by investing in shares of company A are very bright’, etc.
Each of the given sentences involves an element of uncertainty.
A phenomenon or an experiment which can result into more than one possible outcome, is
called a random phenomenon or random experiment or statistical experiment. Although, we
may be aware of all the possible outcomes of a random experiment, it is not possible to
predetermine the outcome associated with a particular experimentation or trial.
Consider, for example, the toss of a coin. The result of a toss can be a head or a tail, therefore, it
is a random experiment. Here we know that either a head or a tail would occur as a result of the
toss, however, it is not possible to predetermine the outcome. With the use of probability
theory, it is possible to assign a quantitative measure, to express the extent of uncertainty,
associated with the occurrence of each possible outcome of a random experiment.

Did u know? The concept of probability originated from the analysis of the games of
chance in the 17th century.

12.1 Definitions

Classical Definition: This definition, also known as the mathematical definition of probability,
was given by J. Bernoulli. With the use of this definition, the probabilities associated with the
occurrence of various events are determined by specifying the conditions of a random
experiment.

Did u know? The classical definition of probability is also known as ‘a priori’ definition of
probability.
Definition
If n is the number of equally likely, mutually exclusive and exhaustive outcomes of a random
experiment out of which m outcomes are favourable to the occurrence of an event A, then the
probability that A occurs, denoted by P(A), is given by:

Number of outcomes favourable to A m
P A
Number of exhaustive outcomes n
Various terms used in the above definition are explained below:
1. Equally likely outcomes: The outcomes of random experiment are said to be equally
likely or equally probable if the occurrence of none of them is expected in preference to
others. For example, if an unbiased coin is tossed, the two possible outcomes, a head or a
tail are equally likely.
2. Mutually exclusive outcomes: Two or more outcomes of an experiment are said to be
mutually exclusive if the occurrence of one of them precludes the occurrence of all others
in the same trial i.e. they cannot occur jointly. For example, the two possible outcomes of
toss of a coin are mutually exclusive. Similarly, the occurrences of the numbers 1, 2, 3, 4, 5,
6 in the roll of a six faced die are mutually exclusive.
3. Exhaustive outcomes: It is the totality of all possible outcomes of a random experiment.
The number of exhaustive outcomes in the roll of a die are six. Similarly, there are
52 exhaustive outcomes in the experiment of drawing a card from a pack of 52 cards.

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