IB Maths: Conditional Probability (AA HL) — FAQ

Worked example: A bag has 5 red, 3 blue balls. Two drawn without replacement. Find P(2nd red | 1st red).

1. P(1st red) = 5/8
2. Given 1st was red, 4 red and 3 blue remain (7 total)
3. P(2nd red | 1st red) = 4/7

Quick tip: whatever comes after the "|" already happened — condition on it, don't recalculate it.

Rearranged, this also gives the multiplication rule P(A∩B) = P(A|B) × P(B), which is what you use to fill in tree diagram branches when the second-stage probability depends on the first.

Test for independence: check whether P(A∩B) = P(A) × P(B). If true, they're independent; if not, you must use the full conditional formula. I've marked scripts where students assumed independence to skip a step — examiners give zero for that assumption unless it's stated or proven.

Steps:

1. Draw first-stage branches with their probabilities.
2. Draw second-stage branches conditional on each first outcome (probabilities must sum to 1 per branch).
3. Multiply along a path for P(A∩B).
4. To find P(A|B), sum all paths giving B, then isolate the path(s) giving A∩B and divide.

Common mistake checklist before you submit:

1. Have you confirmed which event is the given/condition?
2. Does "without replacement" change your denominator on the second draw?
3. Did you use P(A∩B), not P(A) × P(B), unless independence is proven?
4. Have you checked P(B) ≠ 0 before dividing?

Worked example: A test is 95% accurate for a disease with 1% prevalence. Find P(disease | positive test).

Using P(D|+) = P(+|D)P(D) / [P(+|D)P(D) + P(+|D')P(D')] = (0.95×0.01) / (0.95×0.01 + 0.05×0.99) = 0.0095 / 0.059 ≈ 0.161

This surprises most students — a positive result still means only a 16% chance of actually having the disease, because the disease is rare.

On Paper 3, expect the conditional probability part to build on earlier sub-parts — get an early answer wrong and it can cascade, so IB examiners award follow-through marks where your method is consistent even if a prior value was off.

Three habits I see in every 7-scoring student:

1. They write the formula before substituting — examiners award method marks for this even if the final number is wrong.
2. They double-check "with/without replacement" language every single time.
3. They sanity-check answers — a probability above 1 or below 0 means a setup error, and they catch it before submitting.

On RevisionPrep, the Mathematics AA HL Topical Worksheets isolate conditional probability from the rest of Topic 4, so you can drill the exact skill before mixing it back into full-paper timing practice.

Neither route makes the topic itself harder — the difference is how it's assessed. If your child finds symbolic manipulation daunting but is confident with structured real-context data, AI's approach may suit them better, though switching routes purely for one subtopic rarely makes sense given how much else differs between the two.

If your child is aiming for a quantitative degree — economics, engineering, computer science — some university interviews (particularly in the UK) do probe probability reasoning directly, so it's worth being genuinely solid here, not just exam-ready.