Bayes’ Theorem: The Probability Connector

by Yuvi K - December 16, 2023

Bayes’ Theorem: The Probability Connector

The mathematical connection between cause and effect is described in the concept of probability. This concept has given rise to Bayes’ Theorem (originally due to Thomas Bayes), a probability theorem which distinguishes the cause from the effect. Bahrain’s theorem is used to calculate the probability of an event occurring knowing its relation to various other earlier events, or to calculate the probability of a many such events which can produce the given result.

What is Bayes’ Theorem:

Bayes’ Theorem (alternatively, Bayes’ rule or Bayesian theorem, also known as the law of total probability) is a theorem used to calculate the conditional probability of an event based on its relation to other events. It is particularly useful for predicting the probability that an event will occur, given a set of known events and conditions. The theorem is named after Thomas Bayes, an 18th century mathematician who developed it.

Formula of Bayes’ Theorem:

The theorem is commonly expressed as a formula:

P(A|B) = P(B|A)P(A) / P(B)

Where P(A|B) is the conditional probability that event A will occur given that event B has already occurred, P(B|A) is the probability that event B will occur given that event A has already occurred, P(A) is the probability that event A will occur and P(B) is the probability that event B will occur.

Example of Bayes’ Theorem:

To better understand Bayes’ theorem, let’s look at a simple example. Consider a machine which randomly selects balls with colors: green, yellow, and red. The probability that the machine will select a green ball is 45%, yellow ball is 35% and red ball is 20%.

Let’s now assume that we know that a green ball has been selected. What is the probability of the machine selecting a yellow ball on the next draw?

Using Bayes’ theorem, we can calculate the probability of the machine selecting a yellow ball given that a green ball has already been selected.

Here, P(A) is the probability that a yellow ball will be selected, P(B) is the probability that a green ball will be selected and P(B|A) is the probability that a green ball will be selected given that a yellow ball has already been selected.

P(A|B) = (P(B|A)P(A))/P(B)

P(A|B) = (0.2×0.35)/0.45

P(A|B) = 0.22

Therefore, the probability of selecting a yellow ball given that a green ball has already been selected is 0.22.

Uses of Bayes’ Theorem:

Bayes’ theorem is used in a variety of fields, such as in medicine, to determine the probability of certain illnesses. It is also used in artificial intelligence and machine learning, where probabilities are used to make predictions. The theorem can also be used to calculate the likelihood of a set of events occurring given a set of known conditions and events. It is also used in statistics, where it is used to calculate the probability of certain events, given the chance that they could happen.

Conclusion:

Bayes’ theorem is a powerful tool for calculating probabilities and predicting outcomes. It is used in a variety of fields, including medicine, artificial intelligence, and statistics. It can also be used to determine the probability of a given set of events occurring, given the set of conditions and known events.