10-8
10-8
Geometric Probability
1. Plan
Objectives
1
To use segment and area
models to find the
probabilities of events
Examples
1
2
3
4
Finding Probability Using
Segments
Real-World Connection
Finding Probability Using Area
Real-World Connection
What You’ll Learn
Check Skills You’ll Need
• To use segment and area
Find and simplify each ratio.
models to find the
probabilities of events
0
1
C D
2
3
4
E F
5
6
7
8
9 10
6. an odd number 21
5. 4 16
8. a prime number 12
1
3
New Vocabulary • geometric probability
1
Using Segment and Area Models
You may recall that the probability of an event is the ratio of the number of
favorable outcomes to the number of possible outcomes.
Vocabulary Tip
P(event) is read
”the probability
of an event.“
outcomes
P(event) = favorable
possible outcomes
Sometimes you can use a geometric probability model in which you let points
represent outcomes. You find probabilities by comparing measurements of sets of
points. For example, if points of segments represent outcomes, then
Lesson Planning and
Resources
See p. 530E for a list of the
resources that support this lesson.
P(event) =
1
PowerPoint
Bell Ringer Practice
length of favorable segment
length of entire segment .
Finding Probability Using Segments
EXAMPLE
A gnat lands at a random point on the ruler’s edge. Find the probability that the
point is between 3 and 7.
Check Skills You’ll Need
For intervention, direct students to:
Skills Handbook, pp. 756, 762
1
2
3
P(landing between 3 and 7) =
Quick Check
582
4
5
6
7
8
9
10
11
length of favorable segment
4
1
length of entire segment = 12, or 3
1 A point on AB is selected at random.
What is the probability that it is a
point on CD? 25
A
0
C
1
2
3
4
5
6
7
D
B
8
9 10
Chapter 10 Area
Special Needs
Below Level
L1
Have students create a game board as described in
Example 4. Have each group toss a quarter on the
square 100 times. Then find the experimental
probability that it will land in the circle and
compare results.
582
B
You roll a number cube. Find the probability of rolling each of the following.
Math Background
More Math Background: p. 530D
Skills Handbook pages 756 and 762
4. Two circles have radii 1 m and 2 m, respectively. What is the simplest form
of the fraction with numerator equal to the area of the smaller circle and
denominator equal to the area of the larger circle? 14
To find the probability of
winning a carnival game,
as in Example 4
7. 2 or 5
The definition of probability is
numerical in nature, but it allows
for geometric consideration.
Geometric probability is defined
to be the ratio of favorable
length, area, or volume to the
entire length, area, or volume.
A
CE 1 3. AB 1
2. AF
2
BC
BD 1
1. AE
3
. . . And Why
GO for Help
learning style: tactile
L2
Have students design “probability boards” in which
regions of different colors are labeled with the
probability that a dart or coin that lands on the board
lands in that region.
learning style: tactile
You can use a segment model to find the probability of how long you will wait for
a bus.
2
EXAMPLE
Real-World
Connection
Guided Instruction
Commuting Elena’s bus runs every 25 minutes. If she arrives at her bus stop at a
random time, what is the probability that she will have to wait at least 10 minutes
for the bus?
B
A
Assume that a stop takes very little time, and let
0 5 10 15 20 25
AB represent the 25 minutes between buses.
Real-World
If Elena arrives at any time between A and C,
she has to wait at least 10 minutes until B.
Connection
If the bus runs on schedule,
Elena’s average wait
(Example 2) will be 12.5 min.
P(waiting at least 10 min) =
length of AC
length of AB
B
A
C
0
5 10 15 20 25
3
= 15
25, or 5
EXAMPLE
area of favorable region
area of entire region
Finding Probability Using Area
12 in.
12 in.
p(32) 2 p(2) 2
area of red region
5p
= 144 < 0.109, or 10.9%
area of square =
122
The probabilities of hitting the blue, yellow, and red regions are about 2.2%, 6.5%,
and 10.9%, respectively.
Quick Check
3 If you change the blue circle as indicated, how does the probability of hitting the
blue circle change? Explain.
a. Double the radius.
b. Triple the radius.
It becomes 19.6%, or
It becomes about 8.7%,
about 9 times greater.
or about 4 times greater.
Lesson 10-8 Geometric Probability
Advanced Learners
EXAMPLE
3
EXAMPLE
Teaching Tip
Additional Examples
p(1) 2
area of blue region
p
P(blue) = area of square =
= 144 < 0.022, or 2.2% Use a calculator.
122
p(2) 2 2 p(1) 2
area of yellow region
3p
P(yellow) =
=
= 144 < 0.065, or 6.5%
area of square
122
P(red) =
Discuss with students the
probability of the gnat landing
exactly on 6. By the text’s
definition, the probability is 0,
since the length of a point is 0.
PowerPoint
Target Game Assume that a dart you throw
will land on the 1-ft square dartboard and is
equally likely to land at any point on the
board. Find the probability of hitting each of
the blue, yellow, and red regions. The radii of
the concentric circles are 1, 2, and 3 inches,
respectively.
For: Probability Activity
Use: Interactive Textbook, 10-8
Connection to
Probability
Have students explain how to
calculate the numerator of each
of the three probabilities shown.
Ask: Why are terms subtracted in
two of the numerators? You are
finding the difference between
two areas.
If the points of a region represent equally-likely outcomes, then you can find
probabilities by comparing areas.
3
EXAMPLE
Remind students that
probabilities can be written as
fractions, decimals, or percents.
2 What is the probability that Elena will have to wait no more than 10 minutes for
the bus? 25
P(event) =
1
2
The probability that Elena will have to wait at least 10 minutes for the bus is 35
or 60%.
Quick Check
2. Teach
583
1 A gnat lands at random on
the edge of the ruler in Example 1.
Find the probability that the
gnat lands on a point between
2 and 10. 23
2 A museum offers a tour every
hour. If Benny arrives at the tour
site at a random time, what is the
probability that he will have to
wait at least 15 min? 34
3 A circle is inscribed in a square
target with 20-cm sides. Find the
probability that a dart landing
randomly within the square does
not land within the circle.
20 cm
about 21.5%
English Language Learners ELL
L4
Suppose Elena’s bus brings her to another stop, where
an express van runs every 15 minutes. What is the
probability that her total waiting time is more than
20 minutes?
learning style: verbal
Help students recognize that two models for
geometric probability are presented in this lesson: the
segment model shown in Examples 1 and 2, and the
area model shown in Examples 3 and 4.
learning style: verbal
583
4
EXAMPLE
Tactile Learners
Suggest that students model the
example by constructing the game
board and measuring quarters
on it. Ask: Where does the
17
fraction 32
come from? Because
the circle has radius 1 in. and a
quarter has radius 15
32 in., the
quarter will land within the circle
17
if it lies within 1 – 15
32 = 32 in. of
the center of the circle.
As Example 3 suggests, you can apply geometric probability to some games.
This can help you decide how easy or difficult it may be to win such games.
8 in.
4
8 in.
1 in.
17 in.
32
Additional Examples
1 in.
a quarter so that it lands entirely
between the two circles below.
Find the probability that this
happens with a quarter of radius
15
32 in. Assume that the quarter is
equally likely to land anywhere
completely inside the large circle.
Connection
Coin Toss To win a prize in a carnival game, you must toss a quarter so that it lands
entirely within the circle as shown at the left. Find the probability of this happening
on one toss. Assume that the center of a tossed quarter is equally likely to land at
any point within the 8-in. square.
of dashed circle
P(quarter landing in circle) = areaarea
of square
=
15 in.
32
4 To win a prize, you must toss
p Q 17
32 R
82
2
< 0.014, or 1.4%
The probability of a quarter landing in the circle is about 1.4%.
Quick Check
EXERCISES
9 in.
Real-World
15
The radius of the circle is 1 in. The radius of a quarter is 32
in. The favorable points
17
are those that are less than 32 in. from the center of the circle. They are the points
within the dashed circle.
8 in.
PowerPoint
EXAMPLE
4 Critical Thinking Suppose you toss 100 quarters. Would you expect to win a prize?
Explain. Yes; theoretically you should win 1.4 times out of 100.
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
12 in.
A
Practice by Example
Example 1
about 32.6%
GO for
Help
Resources
• Daily Notetaking Guide 10-8
(page 582)
1. CH 12
1
2. FG 10
A B
C D
E
F
G H
I
J
0
2
4
5
6
8
9 10
1
3. DJ 35
3
7
4. EI 25
K
5. AK 1
6. Points M and N are on ZB with ZM = 5, NB = 9, and ZB = 20. A point is
3
chosen at random from ZB. What is the probability that the point is on MN? 10
L3
• Daily Notetaking Guide 10-8—
L1
Adapted Instruction
Find the probability that a point
chosen at random from AK is on
the given segment.
Example 2
(page 583)
Closure
7. Transportation A rapid transit line runs trains every 10 minutes. Draw a
geometric model and find the probability that randomly arriving passengers
will not have to wait more than 4 minutes. See margin.
Traffic Patterns Main Street intersects each street below. The traffic lights on Main
follow the cycles shown. As you travel along Main and approach the intersection,
what is the probability that the first color you see is green?
A target with diameter 16 in. is
formed by two concentric circles.
Assume that a dart is equally
likely to land at any point on the
target, and the probability of
landing in either region is
50 percent. Find the radius of the
smaller circle. Round to the
nearest tenth. 5.7 in.
8. Durham Avenue: green 30 s, yellow 5 s, red 25 s 21 or 50%
4
or about 27%
9. Martin Luther King Boulevard: green 20 s, yellow 5 s, red 50 s 15
10. Yonge Street: green 40 s, yellow 5 s, red 25 s 47 or about 57%
11. International Drive: green 25 s, yellow 5 s, red 45 s 13 or about 33%
7
12. Tamiami Trail: green 35 s, yellow 8 s, red 32 s 15
or about 47%
13. Flutie Pass: green 50 s, yellow 4 s, red 26 s 58 or 62.5%
584
Chapter 10 Area
7. 25 or 40%
0 1 2 3 4 5 6 7 8 9 10
584
14. During May, a certain drawbridge over the Intracoastal Waterway is raised
every half hour to allow boats to pass. It remains open for 5 min. What is the
probability that a motorist arriving at the bridge in May will find it raised? 16
Examples 3, 4
(pages 583 and 584)
Target Games Darts are thrown at each of the boards shown below. A dart hits the
board at a random point. Judging by appearances, find the probability that it will
land in the shaded region.
15.
1
4
or 25%
25%
16.
2
5
17.
3. Practice
Assignment Guide
1 A B 1-45
C Challenge
46-47
Test Prep
Mixed Review
48-51
52-57
or 40%
Homework Quick Check
18.
19.
To check students’ understanding
of key skills and concepts, go over
Exercises 6, 16, 28, 30, 45.
20.
120⬚
2
3
Real-World
Connection
An archer receives from 1 to
10 points for an arrow that
hits the target. A hit in the
center zone is worth 10 points.
B
Apply Your Skills
or about 67%
π
2 1 π
or about 61%
21. Archery An archery target with a radius of
61 cm has 5 scoring zones formed by concentric
circles. The colors of the zones are yellow, red,
blue, black, and white. The radius of the yellow
circle is 12.2 cm. The width of each ring is also
12.2 cm. If an arrow hits the target at a random
point, what is the probability that it hits the
center yellow zone? 4%
42π
4
Error Prevention!
or about 21%
Visual Learners
Exercises 15–17, 19, 20 Discuss
22. BZ contains MN and BZ = 20. A point is chosen at random from BZ. The
probability that the point is also on MN is 0.3, or 30%. Find MN. 6
as a class how to solve these
exercises, which have no
measurements given.
Target Games A dart hits each square dartboard at a random point. Find the
probability that the dart lands inside a circle. Leave your answer in terms of π.
Diversity
π
4
23.
6 cm
π
4
24.
6 cm
π
4
25.
28. Commuting Suppose a bus arrives at a bus stop every 25 min and waits 5 min
GPS before leaving. Sketch a geometric model. Use it to find the probability that a
person has to wait more than 10 min for a bus to leave. See margin.
nline
Exercise 21 Men’s and women’s
archery are Olympic sports. Ask
students who are familiar with
archery to explain the sport.
6 cm
26. A dartboard is a square of radius 10 in. You throw a dart and hit the target.
Find the probability that the dart lies within !10 in. of the center of the square.
10π
200 or about 16%
27. Critical Thinking Use the information given in Example 4.
a. For each 1000 quarters tossed, about how many prizes would be won?14 prizes
b. Suppose the game prize costs the carnival $10. About how much profit would
the carnival expect for every 1000 quarters tossed? $110
GO
Exercise 7 For students confused
by the phrase not . . . more than
4 minutes, ask: What is another
way to state the waiting time?
4 min or less
GPS Guided Problem Solving
Visit: PHSchool.com
Web Code: aue-1008
L4
Enrichment
L2
Reteaching
L1
Adapted Practice
Practice
Name
Class
L3
Date
Practice 10-8
Areas and Volumes of Similar Solids
The figures in each pair are similar. Use the given information to find the
similarity ratio of the smaller figure to the larger figure.
1.
29. Traffic Patterns The traffic lights at Fourth and Commercial Streets repeat
themselves in 60-second cycles. Ms. Li regularly has students drive on Fourth
Street through the Commercial Street intersection. By experience, she knows
that they will face a red light 60% of the time. Use this information to estimate
how long the Fourth Street light is red during each 1-min cycle. 36 s
Homework Help
L3
2.
S.A. = 49 cm2
S.A. = 81 cm2
V = 125 in.3
V = 512 in.3
Are the two solids in each pair similar? If so, give the similarity ratio. If not,
write not similar.
3.
4.
7 in.
8m
14 in.
10.5 in.
6m
4 in.
8 in.
5.
6 in.
3m
4m
6.
9 ft
5 ft
Lesson 10-8 Geometric Probability
585
20 cm
12 ft
15 cm
9 ft
© Pearson Education, Inc. All rights reserved.
12 cm
28. 35 or 60%
wait > 10 min
16 cm
The surface areas of two similar figures are given. The volume of the larger
figure is given. Find the volume of the smaller figure.
7. S.A. = 25 cm2
S.A. = 36 cm2
V = 216 cm3
8. S.A. = 16 in.2
S.A. = 25 in.2
V = 500 in.3
9. S.A. = 72 ft2
S.A. = 98 ft2
V = 686 ft3
The volumes of two similar figures are given. The surface area of the smaller
figure is given. Find the surface area of the larger figure.
10. V = 8 ft3
V = 125 ft3
S.A. = 4 ft2
11. V = 40 m3
V = 135 m3
S.A. = 40 m2
12. V = 125 cm3
V = 1000 cm3
S.A. = 150 cm2
13. A cone-shaped pile of sand weighs 250 lb. How much does a similarly
shaped pile of sand weigh if each dimension is six times as large?
0
Bus
leaves
5
10
15
20
25
Bus
leaves
14. A block of ice weighs 2 lb. How much does a similarly shaped block of
ice weigh if each dimension is twice as large?
585
4. Assess & Reteach
For Exercises 30 and 31, sketch a geometric model and solve.
30. Astronomy Meteoroids (mostly dust-particle size) are continually bombarding
Earth. The surface area of Earth is about 65.7 million square miles. The area of
the United States is about 3.7 million square miles. What is the probability that
a meteoroid landing on Earth will land in the United States? See back of book.
PowerPoint
Lesson Quiz
31. Tape Recording Amy made a tape recording of a chorus rehearsal. The
recording began 21 min into the 60-min tape and lasted 8 min. Later she
accidentally erased a 15-min segment somewhere on the tape. a–b. See margin.
a. In your model show the possible starting times of the erasure. Explain how
you know that the erasure did not start after the 45-min mark.
b. In your model show the starting times of the erasures that would erase the
entire rehearsal. Find the probability that the entire rehearsal was erased.
1. A point on AF is chosen
at random. What is the
probability that it is a
point on BE?
A
B
C
D
E
F
6
7
8
9
10
11
3
5
2. Express elevators to the
top of a tall building leave the
ground floor every 40 seconds.
What is the probability that a
person would have to wait
more than 30 seconds for an
express elevator? 14
Problem Solving Hint
x 2 Algebra Find the probability
0 # P(event) # 1
that coordinate x of a point chosen
at random from AK satisfies
the inequality.
P(event) = 0 means the
event will not occur.
P(event) = 1 means the
event will occur.
32. 2 # x # 8 53
3
33. x $ 7 10
A B
C D
E
F
G H
I
J
0
2
4
5
6
8
9 10
1
3
1
36. 2 # 4x # 3 40
3. regular octagon
40. Background (at right): a circle 40 cm across
Target: a circle with 10-cm radius about 46%
10 cm
41. Background: a square with 40-cm sides
Target: a circle with 10-cm radius about 36%
Real-World
Connection
A mere touch of the target by
the ball triggers the dunk.
1 , or 50%
2
5. circle
Challenge
8 , or 32%
25
Alternative Assessment
586
42. Background: a circle 40-cm across
43. Background: a square with 40-cm sides
Target: a square with 20-cm sides
Target: a square with 20-cm sides
about 58%
about 46%
44. Kimi has a 4-in. straw and a 6-in. straw. She wants to cut the 6-in. straw into two
pieces so that the three pieces form a triangle.
a. If she cuts the straw to get two 3-in. pieces, can she form a triangle? yes
b. If the two pieces are 1 in. and 5 in., can she form a triangle? no
c. If Kimi cuts the straw at a random point, what is the probability that she can
form a triangle? 23
45. a. Open-Ended Design a dartboard game to be used at a charity fair. Specify
the size and shape of the regions of the board. Check students’ work.
b. Writing Describe the rules for using your dartboard and the prizes that
winners receive. Explain how much money you would expect to raise if the
game were played 100 times. Check students’ work.
C
Have teams of four students work
together to design a game that
uses geometric probability. They
should produce a working model
of the game, a set of rules, and
the calculations for all the probabilities involved.
35. 21 x - 5 $ 0 0
9
34. 2x # 9 20
3
37. 0 # 13x + 1 # 5 38. Δx - 6« # 1.5 10
39. Á 2 # px # Á 10
1
See left.
Dunk Tank At a fund-raiser, a volunteer sits on a platform above a tank of water.
She gets dunked when you throw a ball and hit the red target. The radius of the
ball is 3.6 cm. What is the probability that a ball
40 cm
heading randomly for the given background shape
would hit the given target shape?
2 "2
39. "1010π
N 0.06
A dart you throw is equally likely
to land at any point on each
board shown. For Exercises 3–5,
find the probability of its landing
in the shaded area.
3 , or 37.5%
8
4. square
7
K
586
46. Graphing Calculator A circular dartboard has radius 1 m and a yellow circle in
the center. Assume you hit the target at a random point. For what radius of
the yellow center region would P(hitting yellow) equal each of the following?
Use the table feature of a calculator to generate all six answers. Round to the
nearest centimeter.
a. 0.2 45 cm
b. 0.4 63 cm
c. 0.5 71 cm
d. 0.6 77 cm
e. 0.8 89 cm
f. 1.0 100 cm
Chapter 10 Area
31. a.
14 21
b.
0
10
20
30
40
50
60
If it starts after 45 min, you cannot
erase 15 min of a 60 min tape.
0
7
45
10
20
30
or about 16%
40
50
60
Test Prep
47. Target Game A target has a central circle
and three concentric rings. The diameters of
the circles are 2 cm, 6 cm, 10 cm, and 14 cm.
Find the probability of landing in the gray
region. Compare it with the probability of
landing in either the blue or red region.
24 N 49%; the probability is the same.
49
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 593
• Test-Taking Strategies, p. 588
• Test-Taking Strategies with
Transparencies
Test Prep
Multiple Choice
48. A dart hits the dartboard shown. Find the
probability that it lands in the shaded region. A
A. 21%
B. 25%
C. 50%
D. 79%
49. A dart hits the dartboard shown. Find the
probability that it lands in a circle. J
F. 21%
G. 25%
H. 50%
J. 79%
Short Response
Extended Response
4m
50. On this dartboard, the circle with 1-m radius is
inscribed in an equilateral triangle. Find the
probability that a dart that hits the board lands in
the circular region. Justify your answer. See margin.
1m
π(1)2
area of circle
50. [2] area
of triangle ≠ 3 3
N 0.6 ≠ 60%
51. The radius of a circle is 28 m. The measure of the
central angle is 120. 51a–b. See margin.
a. Find the area of the sector in terms of p.
Justify your answer.
b. Find the area of the shaded segment to the
nearest tenth. Justify your answer.
[1] no work shown OR
correct explanation
and a computational
error
120⬚
28 m
51. [4] a. 31 (area of circle) ≠
1
3
GO for
Help
52. A circle has circumference 20p ft. What is its area? 100π ft2
53. A circle has radius 12 cm. What is the area of a sector of the circle with a 308
central angle? 12π cm2
54. What is the area of a semicircle with diameter 20 ft? 50π ft2
Lesson 6-2
x 2 Algebra Find the values of the variables in each parallelogram.
55.
y⬚
4x⬚
Lesson 5-1
57a. D(3, 1); E(1, 4)
56.
x⬚
x ≠ 36; y ≠ 144
(y + x)⬚
x⬚
x ≠ 45; y ≠ 90
3x⬚
57. The coordinates of the vertices of a triangle are A(1, -4), B(5, 6), and C(-3, 2).
a. Find the coordinates of D, the midpoint of AB, and E, the midpoint of BC.
3
3
b. Find the slope of DE and the slope of AC. slope DE ≠ –2 ; slope AC ≠ –2
c. Verify that DE 6 AC. DE and AC have the same slope.
d. Find DE and AC.
e. Verify that DE =
DE ≠ "13 ; AC ≠ 2 "13
m2
≠ 196"3 , so area
of segment ≠
784π – 196 "3 N
3
481.5 m2 ; the
shaded area is the
area of the 1208
sector minus
the area of the k.
[3] one computational
error OR incorrect
explanation
[2] one computational
error and incorrect
explanation
[1] one correct answer
OR a correct
explanation
1 AC. "13 ≠ 1 ? 2 "13
2
2
Lesson 10-8 Geometric Probability
784π
3
b. Area of k ≠
1
1
2 bh ≠ 2 (28"3 )(14)
Mixed Review
Lesson 10-7
(π ? 282) ≠
587
587