Various approaches for calculating Probability

The various approaches for calculating probability are as follows.

1. Theoretical / Classical Approach
2. Frequentist Approach
3. Bayesian Approach

Let us now try to understand the theoretical, frequentist and Bayesian approach with simple examples for each.

1. Theoretical / Classical Approach:

The theoretical approach for probability is defined as the ratio of the number of favourable outcomes to the number of possible outcomes.

Example:

• In an experiment of the flip of a coin, the sample space, in that case, can be defined as S={H, T}.
• Now, let A be the event that we get a Head, so, A={H}.
• In this case, the theoretical approach defines the probability of event A as follows:

Theoretical approach

Here, in the above example,

Theoretical approach

2. Frequentist Approach:

The frequentist approach involves conducting an experiment repeatedly and observing the number of times the event occurs.

Example:

We need to perform the following steps to calculate probability using a frequentist approach.

• Flip the coin n number of times, say 15 times.
• Count the number of times a head appears and let it be denoted by N(A), say 10.
• Calculate the probability of getting a head when you flip a coin can then be calculated as follows:

Frequentist approach

Here, in the above example,

Frequentist approach

3. Bayesian Approach:

Normally, the probability of an event remains constant over time, but in the case of Bayes theorem, a different perspective on calculating the probability is observed. In Bayesian way, the probability of an event changes every time new information with regards to the event arrives. Bayes theorem is an extension of conditional probability as it describes the probability of an event, based on prior knowledge of conditions that might be related to the event.

Bayes’ theorem is also called Bayes’s theorem, Bayes’s law or Bayes’s rule.

Bayesian approach

where,

• P(A|B) is the probability of event A occurring, given event B has occurred
• P(B|A) is the probability of event B occurring, given event A has occurred
• P(A) is the probability of event A
• P(B) is the probability of event B

Here, we need to note that events A and B are independent events (i.e., the probability of the outcome of event A does not depend on the probability of the outcome of event B).

Example:

If we need to find whether the e-mail received is a spam e-mail or not. The event, in this case, is that the e-mail received is spam. The test for spam is that the e-mail contains some flagged words like “lottery”. Here, “The probability an e-mail is spam given that it contains certain flagged words in the content.” We can use Bayes theorem as follows to find whether the e-mail is spam.

Bayes theorem to find whether the e-mail is spam

The applications of Bayes theorem are as follows:

• medicine/ healthcare
• finance
• forecasting
• spam filtering