Probability (Hard)

Why it is harder

Three things the six minute drill does not do. The clock is per question, so you cannot bank time on the easy ones and spend it later; when it hits zero the question is gone. The options are rounded to two significant figures and the exact answer is usually not among them, so you have to estimate and compare rather than recognise. And nothing is marked on screen, so you never know how you are doing until the end. Students who sat the real round after the ramped drill said it had underprepared them; this is the answer to that.

Where we had to choose

Candidate accounts from 2025 and 2026 agree on 90 seconds a question, five options, pick the closest, +1, 1 off and 0 for a skip, and most put the count at 30. Earlier accounts say 20, and a few describe a single 15 or 30 minute clock. We went with the most recent and most detailed reports; Optiver publishes none of it. The questions are our own, written in the families those accounts describe, and the exact answer is shown in the review so you can see how far the closest option was from it.

How to sit it

Guessing is not free: a blind pick from five options is worth minus 0.6 on average. If you can get the shortlist down to two, pick; if not, skip and take the next one. Work in orders of magnitude first, then refine only as far as the gap between the options demands. Most of these questions have a one-line route (Wald, first-step analysis, symmetry, the complement) and the clock is there to find out whether you know it.