Geometric Probability
Overview
Ordinary probability counts outcomes. Geometric probability measures them instead: the probability is the ratio of a favorable measure — a length, an angle measure, a time interval, or an area — to the total measure. It answers questions like the chance that a dropped object lands on a target, or that a song has already played by the time you turn on the radio.
Vocabulary
- Probability
- A numerical value representing the likeliness of an event, expressed as a fraction, a decimal from 0 to 1, or a percent, where 0 means the event will definitely not occur and 1 means it definitely will.
- Geometric Probability
- A probability based on relative measures — length, time, distance, angle measure, or area: p = (measure of the desired outcome) ÷ (measure of the total possible outcomes).
- Favorable Measure
- The measure — length, angle measure, time, or area — of the region or interval in which the desired outcome occurs. It is the numerator of a geometric probability.
- Total Measure
- The measure of the entire segment, figure, circle, or time interval from which the outcome is chosen. It is the denominator of a geometric probability.
- Point Chosen at Random
- The assumption that a point is equally likely to land anywhere in a figure, so the probability of landing in a region depends only on the region's measure compared to the whole.
Main Ideas
- Choose the measure that matches the situation: lengths for a point on a segment, degrees (out of 360°) for a spinner, minutes for a time window, areas for a target on a board.
- A point dropped at random on a 5 ft by 8 ft board with a 3 ft by 3 ft target: p = 9/40 = 22.5%.
- For a shaded region inside a figure, the desired measure is the shaded area — found by subtraction if the shading is what remains after a piece is removed.
- Unequal spinner sections are handled by their degree measures, not by counting sections.
- Simplify the ratio, and be ready to state it as a fraction, a decimal, or a percent.
Lesson Notes
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Practice Problems
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