← Apply probability concepts
1. Show the method. A bare number earns less than the same number with the tree, table or formula that produced it — and if the number is wrong, visible method is the only thing that can still earn a grade.
2. State your assumptions. Usually independence (one event does not affect another) or randomness / equal likelihood. Say which one you assumed and, where it matters, whether it is reasonable in the context.
Probabilities may come from formulae, a distribution table or graph, counts or proportions, simulation results, or plain written information — the method note applies whichever source you use.
Whenever a question gives you raw counts and invites a conclusion about popularity, risk or prevalence, convert to a proportion of the right total first, and say which total you used.
5 Tree diagrams & risk
⚠⭐ Two things the specification requires on every probability question
The NZQA 2026 specification for 91585 says candidates should “clearly show the method they have used to calculate probabilities and state assumptions made.”1. Show the method. A bare number earns less than the same number with the tree, table or formula that produced it — and if the number is wrong, visible method is the only thing that can still earn a grade.
2. State your assumptions. Usually independence (one event does not affect another) or randomness / equal likelihood. Say which one you assumed and, where it matters, whether it is reasonable in the context.
Probabilities may come from formulae, a distribution table or graph, counts or proportions, simulation results, or plain written information — the method note applies whichever source you use.
⭐ Evaluating somebody’s claim
- These questions give you a claim — a coach, a company, a news article — and ask whether the data supports it. That is a conditional probability question in disguise.
- Work out which conditional they actually need, then compute it. The usual trap is that the claim quotes P(positive test | concussed) when what matters is P(concussed | positive test).
- Answer in context, using the numbers. 2025: “explaining why the data does not support the coach’s claim that the test strongly indicates a concussion.”
- Say explicitly whether the claim is supported, partly supported, or not supported, and quote the probability you calculated as the reason.
A rising count is not a rising proportion
If the number of students playing ultimate frisbee grew, that alone proves nothing — the number playing every other sport may have grown too. As the 2025 schedule puts it, the true proportion among all sports “could be lower than the calculated values.”Whenever a question gives you raw counts and invites a conclusion about popularity, risk or prevalence, convert to a proportion of the right total first, and say which total you used.
Using trees
- Multiply along branches (AND); add between branch-paths (OR).
- Branch probabilities at each split sum to 1.
- Great for “without replacement” problems: the second-stage probabilities change.
Risk & relative risk
Absolute risk P(event) in a group e.g. 3 in 100 = 0.03. Relative risk \mathrm{RR}=\frac{\text{risk (group A)}}{\text{risk (group B)}} RR = 2 means twice the risk, but check the absolute numbers too. Worked example Without replacement A bag has 5 red and 3 blue. Two are drawn without replacement. P(both red)? First red: 5/8. After removing one red, 4 red of 7 remain. Second red: 4/7. P(both red) = 5/8 × 4/7 = 20/56 = 5/14. ✓ P(both red) = 5/14 ≈ 0.357.A fast check on a without-replacement tree
- Have I updated BOTH the numerator and the denominator on the second branch, not just the numerator?
- Does my tree show the method explicitly, not just the final fraction?