Section 9.1

Regions in the Unit Square

Probabilities equal to areas in the unit square or a rectangle: meeting problems, conditional regions, timing windows.

Toolkit

Independent uniform quantities on an interval become coordinates of a point uniformly distributed over a square (or rectangle); any probability about them is the area of the corresponding region divided by the total area.

Key techniques:

Complement via corner triangles: for $|X-Y|<d$ (meeting-type) in $[0,1]^2$ , probability is $1-(1-d)^2$ for $0\le d\le1$ .
“Between” / sign conditions split the square into regions by linear boundaries $X=c$ , $Y=c$ , $X+Y=c$ , $X=Y$ ; compute as sums/differences of triangles and rectangles.
Conditioning on a sub-region preserves uniformity there, so conditional probabilities are ratios of sub-areas, not densities.
Cyclic or periodic timing (e.g. a window catching an event) reduces to a length ratio: $\mathbb{P}=\frac{\text{window length}}{\text{cycle length}}$ , adjusted for overlap at cycle boundaries.
Shifting coordinates (e.g. measuring from a midpoint) can turn a problem into two independent uniforms on a smaller symmetric interval, simplifying the region’s shape.

Problems