Geometric Probability
1. The Bus Stop (Length)
A bus arrives randomly between 0 and {{ s1.T }} minutes. You wait at the stop for {{ s1.W }} minutes.
No matter where you place the wait time, its length is the favorable outcome.
Build the probability fraction:
2. Target Practice (Area)
Throw darts at the board! The total radius is R = {{ s2.R }}. The inner circle radius is r = {{ s2.r }}.
Empirical probability is ~{{ (s2.hits/5).toFixed(2) }}.
Calculate the exact theoretical probability:
Correct! Area is proportional to radius squared.
3. The Meeting Problem
Alice and Bob arrive randomly between 12:00 and 1:00 (60 mins). They meet if they arrive within {{ s3.W }} minutes of each other. The green band is the "meeting" region.
The unshaded triangles form a square of side 60 - {{ s3.W }} = {{ 60 - s3.W }}.
So, unshaded area = {{ (60-s3.W)**2 }}.
4. Final Challenge
A square dartboard of side 2R has an inscribed circle of radius R. What is the probability of landing in the corners?
Favorable Area = Area(Square) - Area(Circle)
Build the final simplified fraction: