In this section, let me introduce certain basic terms and definitions that would help you to understand probability better.
Let me start with Sets, sub-sets and their relevance to understanding probability better.
A set is a collection well defined objects. ‘Well defined’ implies that the definition of the object should be clear so that anyone can say that the object belongs to the given set or does not belong to the given set. Once a set is defined, if we are unable to decide whether a given object belongs to the set or not, then it serves no purpose except leading to confusion. Examples of sets can be
- Set of all fruits. Here, the qualifier is that the object must be a fruit. If it is a fruit, it belongs to the set. Otherwise, it does not belong to the set.
- Set of all students. Here, the qualifier is that the person concerned should be a student.
A random experiment is an experiment in which the final outcome cannot be predicted in advance.
In real life, we come across many random experiments. The human life itself is a random experiment. No one can predict who will die or survive the next moment. A farmer plants several seeds in a garden and tend to the plants in the same way. The amount of water, the fertilizer used and every other thing is uniformly done to all the plants. But the yield varies from plant to plant. Thus farming is an example of a random experiment. A set of students appearing for an examination is another example of a random experiment. In the garden, with a row of plants and each plant having number of flowers, a bee choosing a flower is a random experiment.
The set of all possible outcomes of a random experiment is defined as the sample space and is denoted by ‘S’. Every possible subset of ‘S’ is defined as an event. Events are denoted by A,B,C,D etc. Since the empty set or the null set is also a subset of the sample space ‘S’, it is also an event. The null set is known as the impossible event and is denoted by
The sample space ‘S’ is also a subset of itself. Hence, ‘S’ also describes an event. We call ‘S’ the sure event.
As described earlier, the probability of an impossible event is 0 and the probability of the sure event is 1. For any other event ‘A’, p(A) lies between 0 and 1.