Introduction

In the realm of probability, understanding the distinction between conditional and unconditional probabilities is fundamental to making accurate predictions and informed decisions. These two concepts provide insights into how events are affected by specific conditions and how they behave independently of any conditions. In this chapter, we delve into the nuances of conditional and unconditional probabilities, exploring their differences and practical implications.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It involves adjusting the probability of an event based on additional information or conditions. Mathematically, the conditional probability of event A given event $B$ is denoted as $P(A\mid B)$ and is calculated using the formula:

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$

This formula considers the probability of both events A and B occurring and normalizes it with respect to the probability of event $B$. Conditional probability is crucial for understanding how the likelihood of an event changes based on specific circumstances.

Example: Consider rolling a fair six-sided die. Let event A be getting an odd number (1,3, or 5$)$, and event $B$ be getting a number less than 4 (1, 2, or 3$)$. The conditional probability of rolling an odd number given that the number rolled is less than 4 is $P(A\mid B).$ Using the formula: $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{\frac{2}{6}}{\frac{3}{6}}=\frac{2}{3}$$

Unconditional Probability

Unconditional probability, on the other hand, refers to the probability of an event occurring without any specific conditions or context. It represents the inherent likelihood of an event happening in the absence of additional information. Mathematically, the unconditional probability of event $A$ is denoted as $\mathrm{P}(\mathrm{A})$ and is calculated based on the total number of favorable outcomes divided by the total number of possible outcomes.

Example: In the same context of rolling a fair six-sided die, the unconditional probability of rolling an odd number (event $A$ ) is $P(A)$. Since there are three odd numbers out of six possible outcomes, $P(A)=\frac{3}{6}=\frac{1}{2}$.

Difference between Conditional and Unconditional Probabilities

The key distinction between conditional and unconditional probabilities lies in their context. Conditional probability considers the influence of specific conditions, leading to adjusted probabilities based on available information. Unconditional probability provides a baseline likelihood that remains constant regardless of any additional information.