Before you calculate any probability, you must decide which model fits the situation. This page shows how to make that decision, and the four lessons below practise each part of it.
Why does choosing the model matter so much?
Most probability questions look like routine substitution, but the marks are earned earlier. If you name the model correctly, you have done most of the question.
Here is an original contrast using the same numbers. A machine makes bolts, and each bolt is defective with probability 0.05, independently of the others. A quality check takes 8 bolts.
Routine substitution sees “0.05” and “8” and grabs a formula without asking what X counts. Method selection asks three questions first.
- What is X? The number of defective bolts in the check.
- Is there a fixed number of independent trials, each with two outcomes and the same probability? Yes: 8 bolts, defective or not, p = 0.05.
- So X ~ B(8, 0.05), and “at least one defective” is 1 − P(X = 0) = 1 − 0.95⁸ ≈ 0.3366.
Now change the question: the mass of a bolt is measured, with mean 12.0 g and standard deviation 0.2 g. Nothing is being counted, so the binomial model no longer applies. The right model is the normal distribution, and the first step becomes standardising.
How do the parts of the decision fit together?
The four lessons in this cluster each train one piece of the decision.
| Skill | The question it answers | Lesson |
|---|---|---|
| Testing the conditions | Does repeating an event always give a binomial model? | Why repeated events do not always form a binomial model |
| Choosing a shortcut | Is there a quicker route than adding up every term? | Using the complement for an at-least probability |
| Working backwards | Can I recover a score from a stated percentile? | Recovering a raw score from a normal percentile |
| Checking the answer | Does my calculation match the shaded region? | Reconciling a calculation with the shaded diagram |
Who should start where?
A student who is still unsure what a binomial model needs should begin with recognising binomial conditions in the main chapter. A student who is comfortable with both models but loses marks on setup starts with the first lesson above.
If fractions and decimals slow every line of working, the algebra step repair trainer and the word-problem structure worksheet are useful first stops. When you want mixed questions, use the practice set for this cluster.
Will the pace suit a student who needs to revisit algebra?
Yes, because each lesson keeps the algebra small. The numbers are chosen so that the thinking, not the arithmetic, is the hard part, and any step that needs earlier algebra is shown in full.
For a student who wants a teacher to set the pace, online one-to-one Additional Mathematics tuition works from the questions you get wrong. Return to the full chapter at probability distributions.