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Independence, conditioning, and the intuitions that lie.
the shape of this subject
Probability problems trick even careful students. Brainback trains you to write the sample space before you count.
Because probability rewards the students who do the reps — and punishes the ones who outsource them.
Add Probability as a Learn subject and every problem in these areas routes to the Socratic tutor — attempt-first, hint-laddered, output-guarded.
Attempt → correction → guiding question. No answer. Ever. Verbatim from a beta session.
I used complement: P(at least one) = 1 − P(none)…
What's P(A|B) in words before you compute it?
no answer · one guiding question · your next move
the anti-pattern
Answer machines make probability look like a solved problem. It isn't. The exam room reveals whose reasoning was built and whose was rented.
the recall card
Every wrong attempt and low-scored explain-back becomes a card. SM-2 schedules them — the tricky ones come back tomorrow, the solid ones drift out to next week, then next month.
Recall the core concept in your own words. Then apply it to a fresh example under 2 minutes.
The subject where the shortcut has always been most expensive.
Limits, rates, and shapes of change — earned, not delivered.
Where symbolic manipulation becomes second nature — or doesn't.
See the diagram. Not just look at it.
Unit circle, but also unit intuition.
Vectors are directions, not numbers.
→probability — ready when you are
Attempt-first. Never the answer. Add it as a Learn subject in 30 seconds.