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The binomial distribution

Modelling fixed numbers of independent trials with two outcomes, and computing probabilities.

  • Define and explain The binomial distribution in your own words
  • Use key terms such as Binomial distribution accurately
  • Apply what you have learned to new examples and questions
  • Avoid the common mistakes learners make with this topic

Further statistics moves beyond description to inference: generalising from samples to populations. These are the tools behind clinical trials, opinion polls and quality control.

This lesson focuses on The binomial distribution: modelling fixed numbers of independent trials with two outcomes, and computing probabilities.

Definition: The binomial distribution

Modelling fixed numbers of independent trials with two outcomes, and computing probabilities.

Key ideas

Binomial probabilities

P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ. The conditions matter: fixed n, two outcomes, constant p, independent trials. For n = 5, p = 0.3: P(X = 2) = 10 × 0.09 × 0.343 = 0.3087.

Standardising the normal

Convert X ~ N(μ, σ²) to Z ~ N(0, 1) via z = (x − μ)/σ, then read probabilities from tables. The total area under the curve is 1, and the curve is symmetric about μ, so the probability that X falls below the mean μ is 0.5.

Key term — Binomial distribution: The distribution of the number of successes in n independent trials each with success probability p, written X ~ B(n, p).

Worked example: The binomial distribution

Explain why 'the mean has increased' is a one-tailed alternative hypothesis.

It specifies an increase only, so the critical region lies in a single (upper) tail of the distribution.

Answer: It specifies an increase only, so the critical region lies in a single (upper) tail of the distribution.

Common mistakes
  • Using the binomial model when trials are not independent Check the conditions: without replacement from a small population, p changes each trial and the binomial does not apply.
  • Confusing one-tailed and two-tailed tests A two-tailed alternative ('not equal to') splits the significance level across both tails; using one tail doubles your chance of a false rejection.

Practice

X ~ B(4, 0.5). Find P(X = 0).
All four trials fail.

P = C(4,0) × 0.5⁰ × 0.5⁴ = 0.0625.

X ~ B(6, 0.2). Find P(X = 1).
C(6,1) = 6.

P = 6 × 0.2 × 0.8⁵ = 6 × 0.2 × 0.32768 = 0.393216.

X ~ N(100, 15²). Find the z-score for x = 130.
z = (x − μ)/σ.

z = (130 − 100)/15 = 2.

For X ~ N(μ, σ²), what is P(X greater than μ)?
Think about symmetry.

0.5 — half the area lies above the mean.

Quick check

The binomial distribution — quick check

Which of these best defines "Binomial distribution"?

The distribution of the number of successes in n independent trials each with success probability p, written X ~ B(n, p).

A test at the 5% level gives p-value 0.032. What is the conclusion?

0.032 is less than 0.05, so reject H₀; there is sufficient evidence against the null hypothesis.
Key takeaways
  • The binomial distribution: modelling fixed numbers of independent trials with two outcomes, and computing probabilities.
  • Binomial probabilities: P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ.
  • Normal distribution: A continuous symmetric bell-shaped distribution, fully described by its mean μ and variance σ², written X ~ N(μ, σ²).
  • Watch out for: using the binomial model when trials are not independent
The normal distribution →Back to Grade 12 chaptersRelated resources →