# Introduction

Probability may be called a tool. It is a tool used for prediction. Prediction comes in when there is uncertainty. What probability does is to help you predict when you are dealing with uncertainty.

Hence, I define probability as a measure of certainty in an uncertain world.

A measure in Mathematics represents a number. For example, height is a measure. The height of a person is always expressed in terms of a number. Similarly weight is also a measure. The income is a measure. The unit of measurement may change from a number to another. However, a measure is always a number.

On the other hand, the joy you feel when your mother hugs you with love and affection, or, the pride you feel when your father acknowledges your achievement is just not measurable.

When you submit your answer paper in exam, the answers are measured and you are awarded marks that are again numbers. However, if you submit a blank answer paper, there is nothing to measure. For technical reasons, we award 0 as the measure. This is different from the 0 if every answer submitted is wrong.

Thus we have measurable sets , non – measurable sets, sets of measure zero that we come across in our day to day life.

Probability deals with measurable sets and sets of measure zero.

When I say that probability is a measure, it is obvious that probability is always a number. The question that comes up is “What does probability measure?”

###### As discussed earlier, probability deals with uncertainties. However, what it measures is the ‘certainty’ in an uncertain world.

For example. when we say that the probability of getting a head when you toss a coin is 1/2, we are trying to convey that when a coin is tossed, it can result in two possibilities – a head or a tail. Though there is uncertainty, all we can say with certainty is that if you toss a coin two times, you are certain to get head once. This may not be true if you test it by tossing a coin two times. However, when you toss a coin several times, you are certain to observe that it turns out to be a head, half the time.It is for this reason, we say that probability is a measure of certainty in an uncertain world.

If we accept that probability is a measure, then probability is a number. Does it mean that any number represent probability? The answer is a definite “No”.

In real life, when we talk of certainty and uncertainty, there are two extremes of certainty. One extreme is a definite “No” and the other extreme is a definite ‘Yes”. In between the two extremes is “perhaps”. When the answer is a definite “no”, we know that there is no way it will happen. In other words, the number of ways or possibilities it can happen is 0. Hence, in this extreme case, the probability has a numeric value 0 (zero). In the other extreme, if the answer is a definite “yes”, the probability takes a numeric value 1 (one). When the answer is “perhaps”, it hovers between the two extremes and hence the probability has a numeric value between 0 and 1. So we can define that probability is a measure of certainty in an uncertain world and that its numeric value varies from 0 to 1.

We use ‘p’ to denote the probability measure. where p satisfies the condition

0 ≤  p ≤ 1

How to explain or interpret Probability?

Having obtained a value for probability, the question that comes up is how to explain it or interpret it?

Consider a student coming out of the examination hall and he is asked about his performance in the examination. Let us suppose that he replies that he will get A+ grade with probability ¾. How to interpret his answer?

There are different of interpreting this.

1. He is certain that if 4 different people evaluate his answer sheet, then 3 of them will give him A+ grade.
2. if he is asked to answer the same paper 4 times, 3 times he will get A+ grade.
3. The paper is such, 3 out of every 4 students who answer this paper will get A+ grade.

The important thing to note is that each interpretation gives a measure of certainty and not uncertainty.

At the start of a cricket match, the captain of the team responds to a query that there is a 90% chance that he will win the game. How to interpret this?

Here, the captain predicts a 90% chance for his team to win. In other words, he wants to convey

1. If this question is asked 10 times, he would say I will win 9 times.
2. He says that if the two teams meet 10 times in similar conditions, his team will win 9 times.
3. He says 9 out of 10 people will agree with him when he says his team will win.

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