Page 22 - DMTH404_STATISTICS

Statistics

Notes In fact, the famous mathematician, Karl Pearson, performed this experiment 24000 times. He
found that the relative frequency, which is the number of heads divided by the total nuinber of
tosses, approaches 112 as moFe and more repetitions of the experiment are performed. This is
the same figure which the classical probabilists would assign to the probability of obtaining a
head on the toss of a balanced coin.

Thus, it appears that the probability of an event could be interpreted as the long range relative
frequency with which it occurs. This is called the statistical interpretation or the, ‘frequentist
approach to the interpretation of the probability of an event. This approach has its own difficulties.
We’ll not discuss these here. Apart from these two, there are a few other approaches to the
interpretation of probability. These issues are full of philosophical controversies, which are still
not settled.
We, shall adopt the axiomatic approach formulated by Kolmogorov and treat probabilities as
numbers satisfying certain basic rules. This approach is introduced.
We deal with properties of probabilities of events and their computation. We discuss the important
concept of conditional probability of an event given that another event has occurred. It also
includes the celebrated Bayes’ theorem. We discuss the definition and consequences of the
independence of two or more events. Finally, we talk about the probabilistic structure of
independent repetitions of experiments. After getting familiar with the computation of
probabilities in this,unit, we shall take up the study of probability distributions in the next one.

2.1 Probability : Axiomatic Approach

We have considered a number of examples of random experiments in the last unit. The outcomes
of such experiments cannot be predicted in advance. Nevertheless, we frequently make vague
statements about the chances or probabilities associated with outcomes of random experiments,
Cohsider the following examples of such vague statements :
(i) It is very likely that it would rain today.
(ii) The chance that the Indian team will win this match is very small. ‘
(iii) A person who smokes more than 10 cigarettes a day will most probably developing lung
cancer.
(iv) The chances of my whning the first prize in a lottery are negligible.
(v) The price of sugar would most probably increase next week.

Probability theory attempts to quantify such vague statements about the chances being good or
bad, small or large. To give you an idea of such quantification, we describe two simple random
experiments and associate probabilities with their outcomes.

Example 1:

(i) A balanced coin is tossed. The two possible outcomes are head (H) and tail (T). We associate
probability P{H} = 1/2 to the outcome H and probability P{T} = 1/2 to T.
(ii) A person is selected from a large group of persons and his blood group is determined. It
can be one of the four blood groups O, A, B and AB. One possible assignment of probabilities
to these outcomes is given below

Blood group 0 A 3 AB
Probability 0.34 0.27 0.31 0.08

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