Topics
7. Probability
topic badge
United States of AmericaTennessee
Integrated Math 3

7.07 Geometric probability

Lesson
Practice
Lesson

Concept summary

In general, the probability of an event occuring is: P\left(\text{Event}\right)=\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}

This can be applied to geometric contexts in one, two, or three dimensions.

For lengths: P\left(\text{Event}\right)=\dfrac{\text{Length of favorable segment}}{\text{Total length}}

For areas: P\left(\text{Event}\right)=\dfrac{\text{Area of favorable region}}{\text{Total area}}

For volumes: P\left(\text{Event}\right)=\dfrac{\text{Volume of favorable space}}{\text{Total volume}}

Worked examples

Example 1

A random point is chosen along \overline{AD}.

a

Find the probability that the point is on \overline{AB}

Approach

The favorable length in this case is 3 units long while the total length is 15 units long.

Solution

\displaystyle P\left(\text{On } \overline{AB} \right)\displaystyle =\displaystyle \dfrac{\text{Length of favorable segment}}{\text{Total length}}Formula for probability
\displaystyle P\left(\text{On } \overline{AB} \right)\displaystyle =\displaystyle \dfrac{AB}{AD}Identify favorable and total lengths
\displaystyle P\left(\text{On } \overline{AB} \right)\displaystyle =\displaystyle \dfrac{3}{15}Substitute
\displaystyle P\left(\text{On } \overline{AB} \right)\displaystyle =\displaystyle \dfrac{1}{5}Simplify

Reflection

Probability can be expressed as a decimal, fraction, or percentage, so 0.2, \dfrac{1}{5}, or 20 \% are all acceptable answers.

b

Find the probability that the point is on \overline{AC}

Approach

The favorable length in this case is 3+7=10 units long while the total length is 15 units long.

Solution

\displaystyle P\left(\text{On } \overline{AC} \right)\displaystyle =\displaystyle \dfrac{\text{Length of favorable segment}}{\text{Total length}}Formula for probability
\displaystyle P\left(\text{On } \overline{AC} \right)\displaystyle =\displaystyle \dfrac{AC}{AD}Identify favorable and total lengths
\displaystyle P\left(\text{On } \overline{AB} \right)\displaystyle =\displaystyle \dfrac{10}{15}Substitute
\displaystyle P\left(\text{On } \overline{AB} \right)\displaystyle =\displaystyle \dfrac{2}{3}Simplify

Reflection

Since this favorable length is longer than in

a
, the probability should be greater and \dfrac{2}{3}>\dfrac{1}{5}.

Example 2

The probability of a random point on this image being inside the circle is 0.252. If the rectangle is 24\, \text{ft} \times 13\, \text{ft} , determine the radius of the circle.

Approach

We know that P\left(\text{Inside circle}\right)=\dfrac{A_{\text{Circle}}}{A_{\text{Rectangle}}}, so we substitute what we know and solve for the radius.

Solution

\displaystyle P\left(\text{Inside circle}\right)\displaystyle =\displaystyle \dfrac{A_{\text{Circle}}}{A_{\text{Rectangle}}}Probability formula
\displaystyle 0.252\displaystyle =\displaystyle \dfrac{\pi\cdot r^2}{24\cdot13}Substitute given information
\displaystyle 0.252\displaystyle =\displaystyle \dfrac{\pi\cdot r^2}{312}Evaluate the product
\displaystyle 0.252\left(\dfrac{312}{\pi}\right)\displaystyle =\displaystyle r^2Multiplication and division property of equality
\displaystyle \sqrt{0.252\left(\dfrac{312}{\pi}\right)}\displaystyle =\displaystyle rSquare root both sides
\displaystyle 5.002679\ldots\displaystyle =\displaystyle rEvaluate on calculator

The radius of the circle is 5\, \text{ft}

Reflection

We know that the radius must be positive, so we only take the positive square root when solving.

Outcomes

M3.S.CP.C.7

Calculate probabilities using geometric figures.*

M3.MP1

Make sense of problems and persevere in solving them.

M3.MP2

Reason abstractly and quantitatively.

M3.MP4

Model with mathematics.

M3.MP5

Use appropriate tools strategically.

M3.MP6

Attend to precision.

M3.MP7

Look for and make use of structure.

What is Mathspace

About Mathspace