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CIE A-Level Maths Study Notes

4.4.4 Expectation and Variance of Binomial Distribution

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

Marbles

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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)=6×58=3.75E(X) = 6 \times \frac{5}{8} = 3.75
  • Variance: Var(X)=6×58×38=1.40625\text{Var}(X) = 6 \times \frac{5}{8} \times \frac{3}{8} = 1.40625
Marble Graph

Example 2. True/False Quiz

True or False

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

  • Expectation: E(X)=12×0.5=6 E(X) = 12 \times 0.5 = 6
  • Variance: Var(X)=12×0.5×0.5=3\text{Var}(X) = 12 \times 0.5 \times 0.5 = 3
True/False Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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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.

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