First draw
Write the probability of the first outcome over the original total.
AQA 8300 · P8 · Dependent probability
Probability without replacement: the second denominator
The dependency mark. In a without-replacement draw the second branch has one fewer item in the pool, and the mark is for changing the total, not for the arithmetic that follows.
Reviewed family · Studara-authored items · No grade-uplift claim
The lost mark
One answer. Several places the method can fail.
Studara does not reduce the response to right or wrong. The reviewed contract checks these states independently.
Reduced denominator
Write the second-draw probability over the new total, one fewer than before.
Multiply then add
Multiply along each path and add between paths that satisfy the event.
Why it goes wrong
The diagnosis stays uncertainty-safe.
A wrong response can support a specific measured diagnosis. A blind multiple-choice diagnostic records only a possible cause until later working provides stronger evidence.
01
The error
Keeping the original denominator on the second branch — treating a without-replacement draw as if the first item had been put back.
02
Why it survives revision
The tree diagram still looks complete and correctly shaped, every multiplication along the paths is performed correctly, and the total of the branches can still be checked to 1. Exactly one number in a structurally correct answer is wrong, which is the hardest kind of error to see in your own work.
The worked repair
Repair the smallest broken step.
Teaching is not evidence. This one move is the repair; the proof checks begin after it disappears.
Write the new total on the second branch before writing any numerator.
What happens after the repair
Correction is the start, not the result.
Immediate
Asked again straight away, on a different question
The repair leaves the screen and a new authored item checks the method unassisted. Being taught something is never evidence that it was learned, so this is the first stage that counts at all.
Delayed
Asked again after at least 1 day
A parallel form of the same method, unassisted, with the repair no longer on screen. A retest in the same session does not count.
Transfer
Asked again when the surface changes
The transfer form asks for “at least one”, which cannot be read off a single path — it needs paths added, or the complement taken.
Exam
Asked again under exam conditions
Timed, carrying the mark tariff, in an exam-like response format and aligned to the specification. Any one of those missing and the evidence is refused.
REVIEWED BOUNDARY
This page covers one reviewed family: P8 · Dependent probability. Update a without-replacement tree, then multiply within paths and add between them. Other families within the same GCSE Maths topic are outside this reviewed journey, and Studara does not apply the proof label to material that has not been reviewed. No grade-uplift claim is made anywhere on this site.
Straight answers
Before you use this guide.
What changes on a tree diagram without replacement?
The denominator on the second set of branches drops by one, because the first item was not returned. The numerator also drops on any branch matching the first outcome.
How do I answer an 'at least one' probability question?
Either add every path that contains at least one, or find the probability of none and subtract from 1. The second is usually shorter and less error-prone.
Does Studara guarantee a higher grade?
No. Studara has not run a controlled outcome study and publishes no grade-uplift claim. What has been measured, what is withheld, and the metric that failed its bar are all on the public evidence page.
Turn this method into evidence