Probability, a core concept in mathematics, plays a vital role in understanding the likelihood of various events. The focus is on providing a thorough understanding of these rules, their applications, and problem-solving techniques.

## Probability Basics

**Probability of an Event: P(Event)**= Favourable outcomes / Total outcomes.**Sum of Probabilities:**All possible outcomes' probabilities add up to 1.

## Mutually Exclusive Events

**Definition:**Can't happen at the same time. Example: Can't get heads and tails on one coin flip.**Addition Rule:**P(A or B) = P(A) + P(B) for exclusive events A and B.

### Addition Rule Example

**Task:**Probability of drawing a red card or a queen from a deck.**Steps:**- Red Card Probability: $\frac{26}{52}$ (half the deck).
- Queen Probability: $\frac{4}{52}$ (one per suit).

**Add Probabilities:**$\frac{1}{2} + \frac{4}{52}$.**Result:**Simplify to $\frac{15}{26} \approx 57.69\%$.

## Independent Events

**Definition:**One event doesn't affect the other. Example: Rolling a die and flipping a coin.**Multiplication Rule:**P(A and B) = P(A) × P(B) for independent events.

### Multiplication Rule Example

**Task:**Probability of drawing an ace then a king, without replacement.**Steps:**- Ace Probability: $\frac{4}{52}$.
- King after Ace Probability: $\frac{4}{51}$ (one ace gone).
- Multiply Probabilities: $\left(\frac{4}{52}\right) \times \left(\frac{4}{51}\right)$.

**Result:**≈ $\approx 0.60\%$.

## Choosing the Right Rule

**Mutually Exclusive:**Use Addition Rule.**Independent Events:**Use Multiplication Rule.**Analyze Event Relationship:**Decide based on whether events are exclusive or independent.

## General Addition Formula Example

**Task:**Probability of drawing a heart or an ace.**Steps:**- Heart Probability: $\frac{13}{52}$.
- Ace Probability: $\frac{4}{52}$.
- Subtract Ace of Hearts Overlap: $-\frac{1}{52}$.

**Result:**≈ $\approx 30.77\%$.

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.