Mathematics of Probability

How to calculate Probability?

There are different ways to calculate the probability in a given case. They are called the different approaches to Probability. In essence, there are three different approaches to calculating probability. The three different approaches are

  1. The frequency approach
  2. The classical approach or the mathematical approach.
  3. The axiomatic approach.

The frequency approach is based on past history. If there is a pattern established already, the probability is directly based on the past data. For example,if we are interested in calculating the probability of getting a head, when a coin is tossed, we can get an answer in this method.

Let us toss the coin say a hundred times and count the number of heads and tails observed. It can be say 48 and 52. Based on this, probability of getting a head is worked out as 0.48. Instead of 100 times, if we toss the coin, say a 1000 times. If the number of heads observed is 496 and tails is 503, the probability of getting a head is 0.496 (496/1000). If we keep tossing more and mopre, we can see that probability of getting a head approaches 1/2 or 0.5 Based on this past experience, we now say that probability of getting a head is 1/2 or 0.5.

If we consider the mathematical approach, then it is obvious that when a coin is tossed, there are only two possibilities – either getting a head or getting a tail. The mathematical approach defines probability as p =( cases)/(Total no.of cases). i this case, total no. of cases is 2 and favorable no. of cases is 1. hence the required probability

p = 1/2 or p=0.5. As can be seen, the frequency approach and the mathematical approach, both yield the same answers.

The frequency approach does not always help. How many times an experiment should be repeated cannot be decided objectively. As in the case of tossing a coin, should I repeat the experiment 5000 times or 50000 times before I can be sure is a matter of subjective concern. Hence, the frequency approach is not always the first choice in calculating the probability.

The mathematical approach also has its own deficiency. There are times when the mathematical approach can fail. We calculate probability as a ratio of favorable no of cases to the total no of cases. If the total no of cases is infinite, then mathematical approach fails or results in zero always. If the favorable no of cases is also infinite, then probability is indeterminate.

For example, if we are asked the question, what is the probability that the next student picked up at random is a graduate, how to answer this question?

Here, the population is not well defined. We only know that population is the set of all students (if you look at the question carefully, it reads “the next student picked up at random”). The number of students globally is a huge number running into billions. The population can include any student from L.K.G to the Ph.D. level. In the same way, the favorable no. of cases can also be equally huge. In such cases, the mathematical approach fails.

Consider this question. What is the probability that the sun will rise tomorrow?

Here again, the mathematical approach fails.

Thus, we have instances where neither the frequency approach nor the classical or mathematical approach can help us find an answer. To overcome this failure, we have the third approach to probability and this approach is known as

The axiomatic approach.

The axiomatic approach is based on axioms. An axiom is a statement or property that cannot be proved, but is used in proving other results. For example, it is very difficult to give an exact definition of a point. But without a point, there is no geometry. Thus the point is an axiom in Geometry.

In probability, we use three axioms. These three axioms form the foundation for the probability theory. The three axioms are

  1. For any event ‘A’, the probability of its occurrence denoted by p(A) and the first axiom states that p(A) is always non-negative. In other words, the fundamental assumption is that probability cannot be negative. It is either zero or greater than zero.
  2. If we use the symbol ‘S’ to denote the sure event, then p(S) = 1. Thus, the probability of a sure event (an event that is certain to happen) is always equal to 1.
  3. if ‘A’ and ‘B’ are any two mutually exclusive events, the p(AUB) = p(A) + p(B).

To understand the axioms better, I need to introduce certain terms and explain them first. The basic terms and their meanings and usages are explained in the next section.