The Binomial Distribution is a cornerstone in understanding statistical probability, particularly in contexts of binary outcomes like 'success' or 'failure'. This segment explores the expectation (mean) and variance of a Binomial Distribution, pivotal for evaluating its effectiveness in modeling diverse real-life scenarios.

**Binomial Distribution Overview**

Binomial Distribution, B(n, p), helps predict outcomes in scenarios with two possible results (success or failure) across multiple independent trials.

**Trials:**Conducted n times independently.**Success Probability:**Each trial has a fixed probability of success, p.**Outcomes:**Each trial results in either success or failure.**Expectation (Mean):**The average outcome over many trials, calculated as E(X) = np.**Variance:**The spread of outcomes around the mean, calculated as Var(X) = np(1 - p).

## Examples

### Example 1. Marble Drawing

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Draw 6 marbles from a bag with 5 red out of 8 total marbles. The probability of drawing a red marble (p) = 5/8.

**Expectation:**$E(X) = 6 \times \frac{5}{8} = 3.75$**Variance:**$\text{Var}(X) = 6 \times \frac{5}{8} \times \frac{3}{8} = 1.40625$

### Example 2. True/False Quiz

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Answering 12 true/false questions with a 50% chance of guessing correctly.

**Expectation:**$E(X) = 12 \times 0.5 = 6$**Variance:**$\text{Var}(X) = 12 \times 0.5 \times 0.5 = 3$

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.