Study Guide

Binomial distribution

IB Mathematics Analysis and Approaches Higher LevelΒ· Topic 4: Statistics and Probability, 4.11Β· 12 min read

1. Defining the Binomial Settingβ˜…β˜…β˜†β˜†β˜†β± 3 min

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πŸ“˜ Definition

Binomial Setting

A probabilistic scenario that satisfies all four conditions: fixed number of trials n, independent trials, constant success probability p, exactly two outcomes per trial.

βœ“ Quick check

Identify which of the following scenarios qualify as a valid binomial setting

  1. Drawing 5 cards without replacement from a standard deck and counting aces drawn

    Reveal answer
    No β€”

    Trials are not independent, as sampling without replacement changes the success probability each time.

  2. Rolling a fair 6-sided die 10 times and counting how many times you roll a 6

    Reveal answer
    Yes β€”

    All BINS conditions are satisfied: binary outcome, independent rolls, n=10, p=1/6 constant.

2. Binomial Probability Mass Functionβ˜…β˜…β˜…β˜†β˜†β± 4 min

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P(X=k)=(nk)pk(1βˆ’p)nβˆ’k,k=0,1,2,...,nP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0,1,2,...,n
πŸ“ Worked Example

A fair coin is tossed 7 times. Calculate the probability of obtaining exactly 4 heads.

  1. 1

    Define the distribution: X = number of heads in 7 tosses, so X ~ B(7, 0.5)

  2. 2

    Substitute n=7, k=4, p=0.5 into the PMF

  3. 3
    P(X=4)=(74)(0.5)4(0.5)3P(X=4) = \binom{7}{4} (0.5)^4 (0.5)^{3}
  4. 4
    =35Γ—(0.5)7=35/128β‰ˆ0.2734= 35 \times (0.5)^7 = 35 / 128 β‰ˆ 0.2734

3. Expectation and Variance of Binomial Distributionβ˜…β˜…β˜…β˜†β˜†β± 3 min

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E(X)=np,Var(X)=np(1βˆ’p)\mathbb{E}(X) = np, \quad \text{Var}(X) = np(1-p)
πŸ”¬ Derivation
Goal:

Derive E(X) for binomial distribution

Starting from:

X can be written as the sum of n independent Bernoulli random variables X_i, each with E(X_i) = p

  1. 1

    Linearity of expectation gives E(X) = E(X_1 + X_2 + ... + X_n)

  2. 2

    This expands to sum of individual expectations: E(X) = sum_{i=1}^n E(X_i)

  3. 3
    =βˆ‘i=1np=np= \sum_{i=1}^n p = np
Result:

The expectation of a binomial distribution is simply the product of the number of trials and the success probability.

πŸ“ Worked Example

A factory produces defective components with probability 0.02 per item. For a batch of 500 components, find the expected number of defective items and the variance.

  1. 1

    Define X ~ B(500, 0.02)

  2. 2
    E(X)=500Γ—0.02=10E(X) = 500 \times 0.02 = 10
  3. 3
    Var(X)=500Γ—0.02Γ—0.98=9.8Var(X) = 500 \times 0.02 \times 0.98 = 9.8

4. Cumulative Binomial Probability Calculationsβ˜…β˜…β˜…β˜…β˜†β± 4 min

βœ“ Calculator OK

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πŸ“ Worked Example

X ~ B(12, 0.6). Find P(X β‰₯ 9).

  1. 1

    Rewrite the inequality to match standard CDF input: P(X β‰₯9) = 1 - P(X ≀8)

  2. 2

    Use GDC binomcdf(n=12, p=0.6, k=8) to get P(X ≀8) β‰ˆ 0.7747

  3. 3
    P(Xβ‰₯9)=1βˆ’0.7747β‰ˆ0.225P(X β‰₯9) = 1 - 0.7747 β‰ˆ 0.225

5. Common Pitfalls

Wrong move:

Using the binomial distribution for sampling without replacement

Why:

Trials are not independent, so the BINS conditions are violated

Correct move:

Use the hypergeometric distribution instead for sampling without replacement scenarios

Wrong move:

Calculating P(X β‰₯k) as 1 - P(X ≀k)

Why:

This excludes the term P(X=k), leading to an off-by-one error

Correct move:

Rewrite P(X β‰₯k) as 1 - P(X ≀ k-1) to include all valid terms

Wrong move:

Forgetting to use combinations in the PMF, writing P(X=k) = p^k (1-p)^{n-k}

Why:

This only counts the probability of one specific ordered sequence of k successes, not all possible sequences

Correct move:

Multiply by the binomial coefficient n choose k to account for all permutations of successes and failures

Wrong move:

Using the formula Var(X) = np for binomial variance

Why:

This gives the expected value, not the variance, losing the (1-p) factor

Correct move:

Use the full variance formula Var(X) = np(1-p)

Wrong move:

Rounding intermediate probability values early in multi-step problems

Why:

This introduces cascading rounding errors that make your final answer outside the acceptable IB mark range

Correct move:

Keep full unrounded values in your GDC memory until you reach the final step of the calculation

6. Quick Reference Cheatsheet

Quantity

Formula for X ~ B(n,p)

PMF for exactly k successes

P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}

Cumulative probability up to k

P(X ≀k) = \sum_{i=0}^k \binom{n}{i} p^i (1-p)^{n-i}

Expected value

E(X) = np

Variance

Var(X) = np(1-p)

When this came up on past exams

AI-estimated based on syllabus patterns β€” cross-check with official past papers for accuracy. Use only as revision-focus signals.

  • 2023 Β· Paper 2

    Cumulative binomial probability calculation

  • 2022 Β· Paper 3

    Binomial hypothesis test setup

  • 2021 Β· Paper 1

    Solve for n given binomial constraint

What's Next

Mastering binomial distributions is a critical foundation for later topics in this unit, including hypothesis testing for a population proportion, normal approximation to the binomial distribution, and the Poisson distribution for counting events over a fixed interval. You will frequently combine binomial probability rules with combinatorics and conditional probability to solve extended Paper 3 problem sets that carry 15-20% of your total statistics marks. Ensure you can quickly verify if a scenario meets the binomial conditions before selecting this model in exam contexts.