Conditional probability is playing a crucial role in statistics, decision-making, and risk assessment. It focuses on evaluating the probability of one event occurring, given that another event has already taken place. Understanding this interplay between events is vital for a deep comprehension of probability theory.

**Basic Concept**

**Definition:**Conditional probability is the chance of event A happening given that event B has already happened, denoted as P(A|B).**Formula:**$P(A|B) = \frac{P(A \cap B)}{P(B)}$, with P(B) > 0.

## Understanding Events

**Interdependence:**Knowing how one event affects another is crucial.**Sample Spaces:**All possible outcomes in a probability scenario.**Tree Diagrams:**Useful for visualizing probabilities in complex situations.

## Example Scenarios

### A. Dice and Coin Toss:

Image courtesy of wentzwu

**Events:**A = "heads on coin toss", B = "even number on die".**Calculations:**- $P(B) = \frac{3}{6}$ (3 even numbers on die).
- $P(A \text{ and } B) = \frac{1}{2} \times \frac{3}{6} = \frac{1}{4}$ (independent events).
- $P(A|B) = \frac{(1/4)}{(3/6)} = \frac{1}{2}$.

### B. Card Draw from a Deck:

Images courtesy of thoughtsco

**Events:**A = "drawing a king", B = "drawing a face card".**Calculations:**- $P(B) = \frac{12}{52}$ (12 face cards in deck).
- $P(A \text{ and } B) = \frac{4}{52}$ (4 kings, all face cards).
- $P(A|B) = \frac{(4/52)}{(12/52)} = \frac{1}{3}$.

Written by: Dr Rahil Sachak-Patwa

LinkedIn

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.