geometric-probability-area
In a continuous uniform sample space a probability equals the measure of the favorable region divided by the measure of the whole — for planar problems, the favorable area over the total area. The decisive move is translating the stated condition (a comparison, a triangle inequality, an obtuse-angle test) into a planar region bounded by lines and circular arcs, then computing its area as a triangle, a rectangle fraction, or a circular sector plus or minus a triangle.
How to solve
- Keep the unit-square setup but change the condition from y > x (a triangle, P = 3/4) to an arc boundary to force a sector computation
- Scale the two intervals to different lengths so the sample space is a rectangle rather than a square
- Replace a line-bounded favorable region with one bounded by a circular arc to push the difficulty up a grade
Sub-archetype mix (2)
Click a row to see member problems.
- region-area-proportional-probability 25% (3)
Outcomes correspond to sub-regions of a figure and each outcome's probability equals its region's share of the total area; compute the area shares, then combine them (sum, product over independent trials, or parity casework) to answer the question.
- uniform-draw-favorable-region 75% (9)
One or two reals are drawn uniformly from intervals; the condition is translated into a favorable sub-region of the interval, square, or rectangle, and the probability is that region's area over the total — a triangle ratio for line-bounded conditions, a sector plus or minus a triangle for arc-bounded ones.
More data (year-over-year, tool fingerprint, grade distribution, all members)
Tool fingerprint (1–17)
Grade distribution
- Gr 7 7
- Gr 8 5