TutorChase logo
Decorative notebook illustration
CIE A-Level Maths Study Notes

4.5.3 Normal Approximation to the Binomial

Exploring the Normal Approximation to the Binomial Distribution, this section will delve into its conditions, application, and practical problem-solving.

Normal Approximation to Binomial Distribution

1. Conditions for Normal Approximation:

  • np > 5 : Ensures sufficient number of successes.
  • n(1-p) > 5 : Ensures sufficient number of failures.
  • Purpose: To ensure a symmetric distribution suitable for normal approximation.

2. Applying Continuity Correction Factor:

  • For P(Xx)P(X \leq x): Use P(Xx+0.5)P(X \leq x + 0.5).
  • For P(Xx)P(X \geq x): Use P(X > x - 0.5).
  • For P(X=x)P(X = x): Use P(x - 0.5 < X < x + 0.5).
  • Purpose: To align the discrete binomial distribution with the continuous normal distribution for more accurate results.

Examples

Example 1: P(X10)P(X \geq 10) for Binomial Distribution n=30,p=0.2n = 30, p = 0.2

  • Conditions Check: np=6,n(1p)=24np = 6 , n(1-p) = 24 (Both > 5, condition met).
  • Continuity Correction: Adjust to P(X > 9.5).
  • Z-Score Calculation: Z=9.564.81.60Z = \frac{9.5 - 6}{\sqrt{4.8}} \approx 1.60.
  • Probability: P(X10)0.0551P(X \geq 10) \approx 0.0551 (5.51% chance for 10+ successes)
  • Graph:
Binomial Distribution Graph

Example 2: P(3X8)P(3 \leq X \leq 8) for Binomial Distribution n=50,p=0.1n = 50, p = 0.1

  • Conditions Check: np=5,n(1p)=45 np = 5 , n(1-p) = 45 (Both > 5, condition met).
  • Continuity Correction: Adjust to P(2.5X8.5) P(2.5 \leq X \leq 8.5) .
  • Z-Scores: Z1=1.18Z_1 = -1.18 (for X = 2.5), Z2=1.65Z_2 = 1.65 (for X = 8.5).
  • Probability: P(3X8)0.8312P(3 \leq X \leq 8) \approx 0.8312 (83.12% chance for 3-8 successes).
  • Graph:
Binomial Distribution Graph
Dr Rahil Sachak-Patwa avatar
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.

Hire a tutor

Please fill out the form and we'll find a tutor for you.

1/2 About yourself
Still have questions?
Let's get in touch.