Probability Theory: Approaches, Revision, and Random Variables

Different Approaches to Probability Theory

Three different approaches to probability have evolved, mainly to cater to the three different types of situations under which probability measures are normally sought. In this section, we first explore the approaches through examples of distinct types of experiments. The axioms that are common to these approaches are then presented, and the concept of probability is defined using the axioms.

Consider the following situations marked by three distinct types of experiments. The events that we are interested in, within these experiments, are also given.

Situation 1

Experiment: Drawing a number from among nine numbers (say 1 to 9).

Event: On any draw, number 4 occurs.

Situation 2

Experiment: Administering a particular drug.

Event: The drug puts a person to sleep in ten minutes.

Situation 3

Experiment: Commissioning a solar power plant.

Event: The plant turns out to be a successful venture.

The first situation is characterized by the fact that on any draw, each of the nine members has got equal chances of occurrence, if the experiment is conducted with fairness. Thus, in any draw, any one of the numbers may turn up, and the chances of occurrence of each is equal. The probability is calculated as:

number of outcomes favorable to the event / total number of outcomes

Thus, if we denote the event that “a 4 comes out in a draw” as A, and the probability of the event as P(A), then we can see that the total number of outcomes in a draw of numbers 1 to 9 is 9, as any one of the numbers may occur. The number 4 occurs only once in these 9 outcomes. Thus P(A) = 1/9

Revising Probability Estimates

As we have already noted in the introduction, the basic objective behind calculating probabilities is to help us in making decisions. Quite often, whether it is in our personal life or our work life, decision-making is an ongoing process. Consider, for example, a seller of winter garments. You are interested in the demand of the product. To help you in deciding on the amount you should stock for the winter, you have computed the probability of selling different quantities and have noted that the chance of selling a large quantity is very high. Accordingly, you have taken the decision to stock a large quantity of the product. Suppose, when finally the winter comes and the season ends, you discover that you are left with a large quantity of stock. Assuming that you are in this business, you feel that the earlier probability calculation should be updated given the new experience to help you decide on the stock for the next winter. Similarly, situations exist where we are interested in an event on an ongoing basis.

Basic Concepts: Random Variable and Probability Distribution

Before we attempt a formal definition of probability distribution, the concept of ‘random variable' which is central to the theme, needs to be elaborated. In the example given in the Introduction, we have seen that the outcomes of the experiment of a toss of three coins were expressed in terms of the "number of heads." Denoting this "number of heads" by the letter H, we find that in the example, H can assume values of 0, 1, 2, and 3 and corresponding to each value, a probability is associated. This uncertain real variable H, which assumes different numerical values depending on the outcomes of an experiment, and to each of whose values a probability assignment can be made, is known as a random variable. The resulting representation of all the values with their probabilities is termed as the probability distribution of H.

It is customary to present the distribution as follows:

Probability Distribution of Number of Heads (H)

In this case, as we find that H takes only discrete values, the variable H is called a discrete random variable and the resulting distribution is a discrete probability distribution.