1.2.1 Classical Probability

The interpretation of classical probability is often the first one encountered in formal education. This interpretation originated from the games of chance such as rolling dice. Classical probability asserts that in situations where there are multiple possible outcomes, and each outcome is distinct and considered equally likely to occur, the probability is distributed equally among all these outcomes.

Classical Probability is also called “A priori” or “objective” or “Theoretical” probability)

For a random experiment, the probability of an event  can be expressed by the notation:

        

•  is the number of outcomes that can result in the occurrence of .
•  is the total number of possible outcomes in the experiment.

Example 1.1. Consider an experiment that involves flipping a fair coin. Here, we have two equally likely possible outcomes (): heads and tails. Since the coin is fair, both sides have an equal chance of landing. If we want to find the probability of the event “coin landing on heads”, the number of outcomes () where the coin lands on “heads” is just one. Therefore, the probability of a coin landing on heads can be calculated as:

        

Similarly, the probability of the coin landing on tails is also 1/2.

Example 1.2. Consider rolling a fair six-sided die. Here, we have six equally likely outcomes (): numbers 1 to 6. Since it's a fair die, all the numbers have an equal chance of facing up. If we want to find the probability of the event “obtaining any specific number, say 4”, the number of outcomes () in which we can obtain “4” is just one. Hence, the probability of obtaining the number 4 is:

What if we are interested in finding the probability of the event “obtaining an even number”?

In this scenario, the count of outcomes () that result in an even number is three, as there are three even numbers on a die, namely 2, 4, and 6. Therefore, the probability of obtaining an even number can be calculated as follows:

Example 1.3. Now, let's expand upon our example of “coin flip” to include flipping two coins. In this scenario, the two coins can yield four possible outcomes (), as illustrated in Fig 1.1 (The left coin represents the outcome of coin 1, and the right coin represents the outcome of coin 2).

If we were to calculate the “probability of both the coins landing on heads”,  there is only one outcome that can result in the occurrence of the event we are interested in i.e., the outcome where both coins land on heads, shown by Outcome 1 in Fig 1.1. Therefore, the probability of this event can be calculated as:

While classical probability is conceptually straightforward and intuitive, it can be limiting in a lot of complex and real-world situations since the assumption of equally likely outcomes does not hold for them. This is why we need more definitions of probability.