Conditional Probability and Bayes’ Theorem — How New Information Changes Probability

Figure 1. An overview of Conditional Probability, Independent and Dependent Events, the Multiplication Rule, the Law of Total Probability, and Bayes’ Theorem, illustrating how new evidence changes probability and supports decision-making in Machine Learning and AI.

Imagine you are checking the weather before leaving home.

You know that it is cloudy today. Does this mean that the probability of rain is the same as it was yesterday?

Probably not!

The moment you receive new information, your understanding of the situation changes. The probability of rain may increase because cloudy weather provides additional evidence.

This simple idea is the foundation of Conditional Probability and Bayes’ Theorem.

Probability tells us how likely an event is to occur. Conditional probability takes this one step further: it tells us how likely an event is when we already know that another event has occurred.

These concepts are fundamental to statistics and form the mathematical foundation of many Machine Learning, Deep Learning, and Artificial Intelligence algorithms.

Let’s understand them step by step.

1. What Is Conditional Probability?

Suppose we roll a fair six-sided die.

The possible outcomes are:

S = {1, 2, 3, 4, 5, 6}

Let:

• Event A: Getting an even…