← Apply probability concepts
6 Venn diagrams, simulation & counting
Three things that come up repeatedly and are easy marks once you know the shape of the answer.Venn diagrams — usually three circles
- Information arrives as bullet points, and almost always includes overlaps. Always fill the centre first — the “all three” region — then work outwards.
- Each pairwise region is “both minus the centre”. Each single region is “the total for that circle minus everything else already in it”.
- Finish by checking every region sums to the grand total, and remember the region outside all circles: “none of the three”.
- Once filled, any probability is just the count in the region ÷ the relevant total — and for a conditional, the denominator is the given circle, not the grand total.
Working outside-in on a Venn
If you put the “only A” number in first you will almost always double-count, because the totals given usually mean “everyone in circle A including the overlaps”. Centre → pairs → singles → outside, in that order, every time.Choosing a representation
- Two-way table — two categories, and you are given or can find counts. Usually the fastest, and conditionals fall straight out of a row or column.
- Venn diagram — three (sometimes two) overlapping categories described in bullet points.
- Tree diagram — two or more steps of selection or decision. The second layer of branches carries conditional probabilities, P(second | first).
- A two-way table can usually replace a Venn or a tree — if you are stuck on one, try converting.
⭐ Simulation — the answer template
- A simulation imitates the claimed situation so you can see how much the result varies by chance alone.
- The schedule-shaped answer: “A simulation would allow us to see the variation in the proportion of ___ in samples of size ___, based on the assumption that the true proportion is ___%. We could then compare what was observed to the simulated distribution to consider the likelihood of the observed result happening — and so decide whether the reported proportion might be incorrect.”
- Fill in all three blanks with the numbers from the question. The marks are for variation, the assumed true value, the sample size, and the comparison.
- The logic: if the observed result sits far out in the simulated distribution, it is unlikely to have happened by chance, which is evidence against the claim.
Multiplication principle — counting outcomes
- If a task has stages with m then n then p choices, the total number of ways is m × n × p.
- Watch whether choices are used up: picking 3 from 8 people in order gives 8 × 7 × 6, not 8 × 8 × 8.
- If order does not matter, divide by the number of orderings you have counted more than once.
- For small cases, just make a systematic list — it is faster, and a visible list still earns the marks.