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Binomial and normal distributions: A Level Maths knowledge organiser

Everything to know about binomial and normal distributions on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Binomial distribution
B(n, p): the number of successes in n independent trials, each with the same probability p.
Normal distribution
N(μ, σ²): symmetric and bell-shaped, centred on μ.
Standardising
z = (x − μ)/σ turns any normal variable into Z ~ N(0, 1).
Continuity correction
Adjusting by 0.5 when a continuous normal distribution approximates a discrete one.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Binomial \(X\sim B(n,p)\)In the formula booklet: Edexcel, AQA, OCR\(P(X=x)=\tbinom nxp^x(1-p)^{n-x}\)
Binomial mean, varianceIn the formula booklet: Edexcel, AQA, OCR\(E(X)=np,\) \(\mathrm{Var}(X)=np(1-p)\)
Standard normal\(Z=\frac{X-\mu}\sigma\sim N(0,1)\)
Normal: points of inflection at \(\mu\pm\sigma\); about 68%, 95%, 99.7% within 1, 2, 3 s.d.
Normal approx. to \(B(n,p)\) (\(n\) large, \(p\) near ½)\(N\big(np,\ np(1-p)\big)\)
Use a continuity correction, e.g. \(P(X\le4)\approx P(Y<4.5)\)

Worked example

X ~ N(20, 3²). Find P(X > 23).

  1. z = (23 − 20)/3 = 1
  2. P(X > 23) = P(Z > 1) = 1 − 0.8413

Answer: P(X > 23) = 0.159 (3 s.f.)

Common mistakes

  • Binomial probabilities: "at least" and "more than" read wrongly
  • Normal distribution: continuity correction used on a continuous variable
  • P(X > 3) found as 1 − P(X ≤ 4)
  • Conditions for B(n, p) not stated in context

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

  • Write probability distributions for discrete random variables as tables or functions and use the total probability of 1 to find unknown probabilities.
  • Recognise situations that fit the binomial distribution B(n, p), state its conditions and calculate probabilities such as P(X = 3) using the formula.
  • Describe the shape and properties of the normal distribution N(μ, σ²), including symmetry about the mean and the proportion of data within one, two and three standard deviations.
  • Find probabilities for normally distributed variables, such as P(X < 52) and P(45 < X < 60), using a calculator and sketches.
  • Use the standard normal distribution Z = (X − μ)/σ to find an unknown mean or standard deviation, or both, from given probabilities.
  • Approximate a binomial distribution B(n, p) with a normal distribution when n is large and p is near 0.5, using a continuity correction.

The printable sheet

Binomial and normal distributions knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes
Binomial and normal distributions knowledge organiser (A Level Maths), A4. Download the PDF.

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