Skip to main content
MathidiaMathematics Academy
The normal distribution
Lesson 50 of 75
Review

Normal approximation to binomial

Objective. Apply continuity corrections and assess conditions.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The normal approximation to the binomial distribution replaces X ~ B(n,p) with Y ~ N(np, np(1−p)) when n is large and p is not too close to 0 or 1, typically checked using np > 5 and n(1−p) > 5. A continuity correction adjusts for the discrete-to-continuous switch, such as using x+0.5 for P(X ≤ x). For X ~ B(100, 0.5), the approximation is N(50, 25), and P(X ≤ 45) becomes P(Y < 45.5) after correction. Always apply the correct direction of continuity correction based on the inequality in the original probability.

Worked example

Use the normal approximation to binomial for X ~ B(200, 0.4) to estimate P(X ≤ 70), given Φ(−1.09) = 0.1379.

09) = 0
Illustrated card showing 09) = 0
  1. 1Find np = 80 and np(1−p) = 48, so Y ~ N(80, 48)
  2. 2Apply continuity correction: P(X ≤ 70) ≈ P(Y < 70.5)
  3. 3Standardise: z = (70.5−80)/√48 = −1.09
  4. 4Use Φ(−1.09) = 0.1379 as the probability

Answer: 0.1379

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. A common mistake is omitting the continuity correction entirely, using P(Y < 45) instead of P(Y < 45.5) for P(X ≤ 45). Always adjust by 0.5 in the direction that includes the boundary value being approximated.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard