Introduction

In probability, understanding the concepts of independent events and mutually exclusive events is paramount. These concepts lay the foundation for assessing the likelihood of multiple events occurring in various scenarios. In this chapter, we will delve into the intricacies of independent events and mutually exclusive events, exploring their definitions, characteristics, and practical implications.

Probability is a tool that empowers us to make informed decisions in uncertain situations. When dealing with multiple events, we often encounter cases where the occurrence or non-occurrence of one event impacts the likelihood of another event. This leads us to the concepts of independent events and mutually exclusive events.

Independent Events

Two events are considered independent when the occurrence or non-occurrence of one event does not influence the probability of the other event. Mathematically, events A and B are independent if and only if the probability of both events occurring is the product of their individual probabilities:

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

Independent events often arise in scenarios where one event’s outcome doesn’t affect the other event.

Example 1: Consider rolling a fair six-sided die twice. Let event A be getting an even number on the first roll, and event B be getting a number less than 4 on the second roll. These events are independent, as the outcome of the first roll does not affect the outcome of the second roll.

Example 2: Let’s say you are drawing cards from a well-shuffled deck. If you draw a card and then replace it before drawing again, the two draws are independent events.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur simultaneously. If event A occurs, event B cannot occur, and vice versa. In terms of probability, the probability of both mutually exclusive events occurring is zero:

$$P(A \cap B)=0$$

Mutually exclusive events are encountered when outcomes are incompatible. When considering events such as “rainy day” and “sunny day,” they cannot occur simultaneously.

Example: Imagine flipping a coin. Let event A be getting heads, and event B be getting tails. These events are mutually exclusive because it’s impossible to get both heads and tails on a single flip.