Probability, a cornerstone in mathematics, deals with measuring how likely events are to occur. It's crucial in various fields and everyday life. This section explores probability's basic principles, demonstrating their applications in scenarios like dice throws and selecting balls from a bag.

**Basic Principles of Probability**

**Probability of an Event (P(A)):**Ratio of favourable outcomes to total outcomes.

### Key Concepts

1. **Event: **An outcome from a random experiment.

2. **Probability Scale: **0 (impossible) to 1 (certain).

3.** Favourable Outcomes:** Outcomes that match the event criteria.

### Example: Coin Flip

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- Flipping a fair coin.
- Probability of heads $P(H) = \frac{1}{2}$

## Enumerating Outcomes in Equiprobable Cases

- All outcomes are equally likely.
- Techniques: Listing Outcomes, Defining Sample Space.

### Example: Dice Roll

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- Rolling a six-sided die.
- Each outcome (1-6) chance = 1/6.

## Permutations and Combinations

**Permutations:**Order matters. Formula: $P(n, r) = \frac{n!}{(n-r)!}$.**Example:**Arranging 3 out of 5 books = 60 ways.

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**Combinations:**Order doesn't matter. Formula: $C(n, r) = \frac{n!}{r! \cdot (n-r)!}$.**Example:**Choosing 3 from 10 people = 120 ways.

## Application in Probability

### 1. Drawing Balls from a Bag

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**Problem:**Probability of 2 red balls from 5 red, 3 blue (no replacement).**Steps:**Calculate total outcomes $C(8, 2) = 28$, favorable outcomes $C(5, 2) = 10$, then probability $\frac{10}{28} \approx 0.357$.

## 2. Dice Throw Sum

**Problem:**Probability of sum 8 when rolling dice twice.**Steps:**Total outcomes (6x6 = 36), favorable outcomes (5 pairs sum to 8), then probability $\frac{5}{36} \approx 0.139$.

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.