geometry unit 10.8

UNIT 10.8 GEOMETRIC PROBABILITYUNIT 10.8 GEOMETRIC PROBABILITY

Warm UpFind the area of each figure.

1. A = 36 ft2

A = 20 m22.

3. 3 points in the figure are chosen randomly. What is the probability that they are collinear?0.2

Calculate geometric probabilities.

Use geometric probability to predict results in real-world situations.

Objectives

geometric probability

Vocabulary

Remember that in probability, the set ofall possible outcomes of an experimentis called the sample space. Any set ofoutcomes is called an event.

If every outcome in the sample space isequally likely, the theoretical probabilityof an event is

Geometric probability is used when an experiment has an infinite number of outcomes. In geometric probability, the probability of an event is based on a ratio of geometric measures such as length or area. The outcomes of anexperiment may be points on a segment or in a plane figure.

If an event has a probability p of occurring, the probability of the event not occurring is 1 – p.

Remember!

A point is chosen randomly on PS. Find the probability of each event.

Example 1A: Using Length to Find Geometric Probability

The point is on RS.

Example 1B: Using Length to Find Geometric Probability

The point is not on QR.

Subtract from 1 to find the probability that the point is not on QR.

Example 1C: Using Length to Find Geometric Probability

The point is on PQ or QR.

P(PQ or QR) = P(PQ) + P(QR)

Check It Out! Example 1

Use the figure below to find the probability that the point is on BD.

A pedestrian signal at a crosswalk has the following cycle: “WALK” for 45 seconds and “DON’T WALK” for 70 seconds.

Example 2A: Transportation Application

What is the probability the signal will show “WALK” when you arrive?

To find the probability, draw a segment to represent the number of seconds that each signal is on.

The signal is “WALK” for 45 out of every 115 seconds.

Example 2B: Transportation Application

If you arrive at the signal 40 times, predict about how many times you will have to stop and wait more than 40 seconds.

In the model, the event of stopping and waiting more than 40 seconds is represented by a segment that starts at B and ends 40 units from C. The probability of stopping and waiting more than 40

seconds is

If you arrive at the light 40 times, you will probably stop and wait more than 40 seconds about (40) ≈ 10 times.

Check It Out! Example 2

Use the information below. What is the probability that the light will not be on red when you arrive?

The probability that the light will be on red is

Use the spinner to find the probability of each event.

Example 3A: Using Angle Measures to Find Geometric Probability

the pointer landing on yellow

The angle measure in the yellow region is 140°.

Example 3B: Using Angle Measures to Find Geometric Probability

the pointer landing on blue or red

The angle measure in the blue region is 52°.

The angle measure in the red region is 60°.

Use the spinner to find the probability of each event.

Example 3C: Using Angle Measures to Find Geometric Probability

the pointer not landing on green

The angle measure in the green region is 108°.

Subtract this angle measure from 360°.

Use the spinner to find the probability of each event.

Check It Out! Example 3

Use the spinner below to find the probability of the pointer landing on red or yellow.

The probability is that the

spinner will land on red or

yellow.

Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth.

Example 4: Using Area to find Geometric Probability

Example 4A: Using Area to find Geometric Probability

the circle

The area of the circle is A = πr2

= π(9)2 = 81π ≈ 254.5 ft2.

The area of the rectangle is A = bh

= 50(28) = 1400 ft2.

The probability is P = 254.51400

≈ 0.18.

Example 4B: Using Area to find Geometric Probability

the trapezoid

The area of the rectangle is A = bh

= 50(28) = 1400 ft2.

The area of the trapezoid is

The probability is

Example 4C: Using Area to find Geometric Probability

one of the two squares

The area of the two squares is A = 2s2

= 2(10)2 = 200 ft2.

The area of the rectangle is A = bh

= 50(28) = 1400 ft2.

The probability is

Check It Out! Example 4

Area of rectangle: 900 m2

Find the probability that a point chosen randomly inside the rectangle is not inside the triangle, circle, or trapezoid. Round to the nearest hundredth.

The probability of landing inside the triangle (and circle) and trapezoid is 0.29.Probability of not landing in these areas is 1 – 0.29 = 0.71.

Lesson Quiz: Part I

A point is chosen randomly on EH. Find the probability of each event.

1. The point is on EG.

2. The point is not on EF.

35

1315

Lesson Quiz: Part II

3. An antivirus program has the following cycle: scan: 15 min, display results: 5 min, sleep: 40 min. Find the probability that the program will be scanning when you arrive at the computer.

0.25

4. Use the spinner to find the probability of the pointer landing on a shaded area.

0.5

Lesson Quiz: Part III

5. Find the probability that a point chosen randomly inside the rectangle is in the triangle.

0.25

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