Introduction

Probability theory offers a deeper understanding of events and their relationships. Two fundamental concepts in probability are independent events and conditionally independent events. In this chapter, we will delve into the distinctions between these concepts, unraveling their implications and applications.

Probability is a powerful tool that helps us quantify uncertainty. In real-life situations, events may or may not influence each other. Understanding the difference between independent and conditionally independent events is essential for making informed decisions based on probabilities.

Independent Events

Two events are considered independent when the occurrence of one event does not affect the probability of the other event happening. In mathematical terms, events A and B are independent if and only if the probability of both events occurring is the product of their individual probabilities:

$$P(A \cap B)=P(A) \times P(B)$$

Example: Consider the toss of a fair coin twice. The outcomes of the two tosses are independent events. The probability of getting heads in the first toss and tails in the second toss is $\frac{1}{2} \times$ $\frac{1}{2}=\frac{1}{4}$, which is also the probability of getting tails in the first toss and heads in the second toss.

Conditionally Independent Events

Conditionally independent events are a bit more nuanced. Two events, A and B, are conditionally independent given a third event C if the occurrence of event C makes events A and B independent of each other. Mathematically, events A and B are conditionally independent given C if:

$$P(A \cap B \mid C)=P(A \mid C) \times P(B \mid C)$$

Example: Imagine rolling two dice. Let event A be the sum of the dice being odd, event B be the sum being greater than 9, and event C be the sum being greater than 5. While events A and B are not independent, they become conditionally independent given event C, because if you know that the sum is greater than 5, it doesn’t influence whether it’s odd or whether it’s over 9.