Page 24 - Probability and Statistical Inference

1

Notions of Probability

1.1 Introduction

In the study of the subject of probability, we first imagine an appropriate
random experiment. A random experiment has three important components
which are:

a) multiplicity of outcomes,
b) uncertainty regarding the outcomes, and
c) repeatability of the experiment in identical fashions.

Suppose that one tosses a regular coin up in the air. The coin has two sides,
namely the head (H) and tail (T). Let us assume that the tossed coin will land
on either H or T. Every time one tosses the coin, there is the possibility of the
coin landing on its head or tail (multiplicity of outcomes). But, no one can say
with absolute certainty whether the coin would land on its head, or for that
matter, on its tail (uncertainty regarding the outcomes). One may toss this coin
as many times as one likes under identical conditions (repeatability) provided
the coin is not damaged in any way in the process of tossing it successively.
All three components are crucial ingredients of a random experiment. In
order to contrast a random experiment with another experiment, suppose that in
a lab environment, a bowl of pure water is boiled and the boiling temperature is
then recorded. The first time this experiment is performed, the recorded tem-
perature would read 100° Celsius (or 212° Fahrenheit). Under identical and
perfect lab conditions, we can think of repeating this experiment several times,
but then each time the boiling temperature would read 100° Celsius (or 212°
Fahrenheit). Such an experiment will not fall in the category of a random ex-
periment because the requirements of multiplicity and uncertainty of the out-
comes are both violated here.
We interpret probability of an event as the relative frequency of the oc-
currence of that event in a number of independent and identical replications
of the experiment. We may be curious to know the magnitude of the prob-
ability p of observing a head (H) when a particular coin is tossed. In order to
gather valuable information about p, we may decide to toss the coin ten times,
for example, and suppose that the following sequence of H and T is