Geometric Probability | Moosa Academy

Geometric Probability — Moosa Academy

When outcomes cannot be counted one by one — a dart landing anywhere on a board, a train arriving at any moment — probability comes from a measure instead. Geometric probability is the favorable measure divided by the total measure.

Definition Part over whole

P = \frac{\text{favorable measure}}{\text{total measure}}

The same idea covers any continuous measure — you just divide the part you want by the whole:

Lengths: favorable length ÷ total length.

Areas: favorable area ÷ total area.

Times: favorable time ÷ total time.

Volumes: favorable volume ÷ total volume.

It assumes a uniform distribution — every point is equally likely.

Example A dart on a target

The board has radius $R = 150$ ; the target has radius $r = 60$ . The $\pi$ cancels:

P = \dfrac{\pi r^{2}}{\pi R^{2}} = \dfrac{60^{2}}{150^{2}} = \dfrac{3600}{22500}

⟹ P = 16%.

Example Three more

Waiting for a train (time)

A train comes every 20 minutes; a wait of up to 5 minutes is fine.

P = \dfrac{5}{20}

⟹ P = 25%.

A pond in a garden (area)

A 300 × 200 garden holds a circular pond of radius 50.

P = \dfrac{\pi \times 50^{2}}{300 \times 200} = \dfrac{7854}{60000}

⟹ P ≈ 13.09%.

Four circles on a square board (area)

A square of side 20 cm holds four circles of radius 3 cm.

P = \dfrac{4 \times \pi \times 3^{2}}{20^{2}} = \dfrac{113.1}{400}

⟹ P ≈ 28.3%.

Reference Four kinds of measure

Type	Formula	Example
Areas	area ÷ total area	a circle inside a rectangle
Lengths	length ÷ total length	a point on a segment
Times	time ÷ total time	train timetables
Volumes	volume ÷ total volume	a ball inside a box

Summary

Geometric probability applies to continuous measures — we do not count discrete outcomes.
The formula is $P = \dfrac{\text{part}}{\text{whole}}$ — it works for length, area, time, and volume.
For circles, area $= \pi r^{2}$ , and the $\pi$ cancels when you divide.
It assumes a uniform distribution: every point is equally likely.