Today is rung 7 of 9 on this topic
Probability, permutations and combinations
Can do: Counts outcomes correctly and calculates conditional probability without falling for the classic traps.
Why it matters
Conditional probability is where intuition fails hardest, and it's exactly the reasoning behind medical tests, risk and any real evidence.
What it looks like when it's wobbly
Confuses permutations with combinations, and reverses conditional probabilities without noticing.
⏱️ Two-minute check
"How many ways to pick 3 from 8 if order doesn't matter?" Then a conditional test-accuracy problem.
Solid looks like: 56, and awareness that a positive test doesn't mean the same as the test's accuracy figure.
The full activity for this topic · 30 minutes
The Test Accuracy Problem
- 1Set up a rare condition — 1 in 1,000 — and a test that's 99% accurate.
- 2Work with 100,000 imaginary people rather than percentages. Natural frequencies make it obvious.
- 3Build the table of true positives, false positives, and totals.
- 4Calculate the chance a positive result actually means the condition. It's shockingly low.
- 5Discuss why this matters for real screening decisions.
If it's too hard
Counting problems with the formulas only.
If it's too easy
Bayes' theorem written formally after doing it with frequencies.
Say this
"Don't use percentages. Take a hundred thousand people and count them — the answer becomes obvious."