Page 78 - DMTH404_STATISTICS

Statistics

Notes 6.1 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. It is because
of this that the classical definition 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.
4. Event: The occurrence or non-occurrence of a phenomenon is called an event. For example,
in the toss of two coins, there are four exhaustive outcomes, viz. (H, H), (H, T), (T, H),
(T, T). The events associated with this experiment can be defined in a number of ways. For
example, (i) the event of occurrence of head on both the coins, (ii) the event of occurrence
of head on at least one of the two coins, (iii) the event of non-occurrence of head on the two
coins, etc.

An event can be simple or composite depending upon whether it corresponds to a single
outcome of the experiment or not. In the example, given above, the event defined by (i) is
simple, while those defined by (ii) and (iii) are composite events.

Example 1: What is the probability of obtaining a head in the toss of an unbiased coin?
Solution.
This experiment has two possible outcomes, i.e., occurrence of a head or tail. These two outcomes
are mutually exclusive and exhaustive. Since the coin is given to be unbiased, the two outcomes
are equally likely. Thus, all the conditions of the classical definition are satisfied.
No. of cases favourable to the occurrence of head= 1

No. of exhaustive cases = 2

1
P H
 Probability of obtaining head ( ) = .
2

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