The binomial distribution
By the end, you can
- Check the assumptions for a binomial model
- Calculate exact-count and cumulative probabilities
- Use the binomial mean and variance
about 25 minutes for the lesson · practise at your own pace.
The central idea
A binomial variable counts successes in n independent trials, each with the same success probability p. For , . Its mean is np and variance np(1−p). A fixed trial count, two categories per trial, constant p and independence are model assumptions.
Worked example
A quality-control model treats each independently produced item as defective with probability 0.2. For a batch of three items, let be the number defective. The count is fixed at three trials, each item is classified as defective or not, and the model assumes independent outcomes with the same defect probability. Thus .
“At least one defective” means , so .
The expected count is and the variance is . The non-integer expectation is an average across batches, not a possible count in one batch.
A fixed trial count and a common marginal success probability do not establish independence. Check for shared causes, changing conditions and sampling without replacement.
Check your understanding
A bag contains four red and six blue counters. Three counters are selected.
In procedure A each counter is replaced and the bag is thoroughly mixed before the next random draw; procedure B keeps each drawn counter out.
Decide which red-count model is binomial and justify the difference.
Find the probability of at least one red counter in each procedure and explain why a common first-draw probability is not enough to use the same model.
For procedure A, the independent draws each have red probability , so and . For procedure B, successive draws are dependent and the conditional red probability changes with previous colours. Compute the complement directly: . Both procedures start with red probability , but only replacement and independent random mixing keep the binomial trial model valid throughout.
Choose a practice question
Mastery checklist: after the lecture and after the tutorial
Use the same five checks at both checkpoints in the printed notes:
- Explain as a discrete probability distribution for a success count.
- Use it as a model and recall all four conditions: fixed trial count, success/failure classification, constant success probability and independent trials.
- Calculate binomial probabilities using the formula or a graphing calculator, with integer endpoints handled correctly.
- Find and use the mean and variance , without needing their proofs in this chapter.
- Find the mode or both modes, distinguishing exact ties from display rounding.
Choose practice by the skill you need
The complete seven-row source map is retained below with links. Part labels and broad groupings follow the printed map; a dash is a blank source cell. Read the qualifications below before using its “number of trials” row.
| Skill/concept | Lecture examples | Discussion | Review problems |
|---|---|---|---|
| Conditions for a suitable binomial model | 4(i), 8(i), 11(i) | 4(i) | 1 (last part), 5(i), 15(i), 16(i), 17(i), 18(i) |
| Point probabilities | 2(i), 3 | 1(v), 2, 3(i), 4(iii), 8(a)(i) | 2, 3(ii), 4(a)(i), 8, 9(i)(ii), 16(iii)(iv) |
| Cumulative probabilities | 2(ii)–(v), 8(ii), 9(ii) | 1, 2, 4(ii), 5 (second part), 7(i), 8(a)(ii), 9(ii)(iii) | 1(ii), 3(i), 4(a)(ii), (b), 5(iii), 6(i), 7, 9(iv), 10(i)(ii), 12, 13(ii), 16(ii), 17(iii), 18(ii)(iii) |
| Given probability: source row “n, number of trials” | 4(ii), 5, 7, 8(iii) | 3(ii), 6 | 5(ii), 6(ii), 9(iii), 10(iii), 11(i), 15(iv), 17(ii) |
| Given probability: observed value x | 6 | 8(c) | — |
| Mean and variance | 9(i) | 5 (first part) | 10 (first part), 11(ii), 12, 13(i), 15(iii) |
| Mode | 10, 11(ii) | 7(ii), 8(b), 9(iv) | 12, 13(iii), 14 |
Use the actual unknown in the question. The source’s n-row also includes probability-p equations: Lecture 7 and 8(iii), Review 5(ii) and 17(ii). Discussion 6’s modified scheme asks for an expression and an expected sample count. They are not all integer-n searches.
Its observed-value row includes Discussion 8(c), a cash cutoff whose event changes at half-dollar steps.
The source map prints a probability formula with failure exponent x; the correct formula used throughout the lesson is . The unchanged source map remains in the retained transcription.
Start with Example 1 if you need to practise the modelling conditions. Then choose a row matching the event or unknown you are working on.
Complete discussion and review sets
Both lecture copies repeat the following questions and short answer keys. The links open the existing worked records so you can try a question before revealing its solution. Every printed ask and mark has been matched to those records.
- Discussion: Question 1, Question 2, Question 3, Question 4, Question 5, Question 6, Question 7, Question 8, Question 9, Question 10.
- Review: Question 1, Question 2, Question 3, Question 4, Question 5, Question 6, Question 7, Question 8, Question 9, Question 10, Question 11, Question 12, Question 13, Question 14, Question 15, Question 16, Question 17, Question 18.
Discussion 9 repeats the label (iii) for its final mode-range ask; its short key and the linked working call that ask (iv). Review 9’s short key labels part(ii) as(b), and Review 11’s key repeats(ii) for its first group; the links follow the actual question parts. Review 9 names August correctly in these lecture copies; its solution PDF misspells it “Agusut”, which is explained in the retained review record.
Practice questions
40 examples and practice questions, in 7 sections.
Ready to try this on your own?
For “The binomial distribution”, try a question without the working and explain your method. This records your confidence, saved in this browser only.