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Geometric Probability

Basic probability counts outcomes — but what if there's nothing to count? A dart can land at infinitely many points on a board, so you can't list a sample space. Geometric probability replaces counting with measuring: the probability is the measure of the region you want divided by the measure of the whole region.

The measure can be a length, an area, or an angle, depending on the situation. A point on a segment uses length, a dart on a board uses area, and a spinner uses the angle of each sector. The fraction is the same idea every time: P=favorable measuretotal measureP = \dfrac{\text{favorable measure}}{\text{total measure}}.

Pick the right measure

Ask what kind of region the random point lives in. If the point lands somewhere on a line segment, compare lengths. If it lands somewhere in a flat region — a dartboard, a floor plan, a target — compare areas. If a spinner arrow stops at a random direction, compare the sector's central angle to 360∘360^\circ.

Then set up the ratio: the measure of the favorable region on top, the measure of the entire region on the bottom. Units cancel, so the answer is a plain number between 00 and 11, just like every probability.

Compare areas, not side lengths

The classic trap: a 33 ft by 33 ft shaded square inside a 99 ft by 99 ft board is not a 39\dfrac{3}{9} chance. The dart lands in two-dimensional space, so the ratio must use areas: 3×39×9=981=19\dfrac{3 \times 3}{9 \times 9} = \dfrac{9}{81} = \dfrac{1}{9}. When each side is 13\dfrac{1}{3} as long, the area — and the probability — is 19\dfrac{1}{9} as big.

The same warning applies to circles. Compare πr2\pi r^2 values, not radii. A circle with half the radius covers one quarter of the area. The figure below shows the shaded square from that trap: its side is one third of the board's, but its area is only one ninth.

99
33

Regions with holes

Sometimes the favorable region is what's left over — the board outside the bullseye, or a ring between two circles. Subtract first: favorable area == big area −- small area, then divide by the total area. The denominator is still the whole region the point can land in.

Worked examples

Example 1: a shaded square on a board

A dart lands at random on a 1010 in by 1010 in square board with a shaded 55 in by 55 in square region. Find the probability the dart lands in the shaded region.

Area of the whole board10×10=10010 \times 10 = 100
Area of the shaded region5×5=255 \times 5 = 25
Probability is the area ratioP=25100=14P = \dfrac{25}{100} = \dfrac{1}{4}

Answer: P=14P = \dfrac{1}{4}

Example 2: a circle inside a circle

A circular splash pad has a radius of 88 ft, and a circular fountain of radius 22 ft sits at its center. A ball lands at random on the pad. Find the probability it lands on the fountain.

Area of the splash padπ(8)2=64π\pi (8)^2 = 64\pi
Area of the fountainπ(2)2=4π\pi (2)^2 = 4\pi
Probability is the area ratio — the π\pi's cancelP=4π64π=116P = \dfrac{4\pi}{64\pi} = \dfrac{1}{16}

Answer: P=116P = \dfrac{1}{16}

Example 3: a spinner sector

A spinner is divided so that a 90∘90^\circ sector is shaded. Find the probability the arrow lands in the shaded sector.

Total angle in the spinner360∘360^\circ
Shaded angle90∘90^\circ
Probability is the angle ratioP=90360=14P = \dfrac{90}{360} = \dfrac{1}{4}

Answer: P=14P = \dfrac{1}{4}

Try one yourself

Common questions

How do I know whether to use length, area, or angle?

Match the measure to where the random point lives. On a segment, use length. In a flat region like a dartboard, use area. On a spinner, use the central angle out of 360∘360^\circ. The formula is the same ratio each time.

Why can't I just compare side lengths or radii?

Because the point lands in a two-dimensional region, and area doesn't scale the same way as length. A square with sides 13\dfrac{1}{3} as long has 19\dfrac{1}{9} the area, so the probability is 19\dfrac{1}{9}, not 13\dfrac{1}{3}.

What if the shaded region is a triangle or an odd shape?

Find its area with the usual formulas — 12bh\dfrac{1}{2}bh for a triangle, or subtract pieces for composite shapes — then divide by the total area. Geometric probability is an area problem wearing a probability costume.

Do the units matter?

Only that they match. If both areas are in square feet, the units cancel and the probability is a pure number. Never mix, say, inches on top with feet on the bottom.

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