Warm-Up Exercises
1. Find the area of a circle with radius 5.2 cm.
ANSWER
84.9 cm2
2. Find the area of a right triangle with hypotenuse 18 in.
and side length 8 in.
ANSWER
64.5 in.2
3. A rectangle with sides 4 inches and 5 inches is cut
out of a square with sides 8 inches. Find the area that
remains.
ANSWER
44in.2
Warm-Up Exercises
Vocabulary
• probability of an event – is a measure of the likelihood
the event will occur;
0 ≤ P(event) ≤ 1
• geometric probability – is a ratio that involves a
geometric measure such as length or area
• probability of an event –
total possible
Warm-Up Exercises
Vocabulary
Geometric Probability
• probability and length –
• probability and area –
Warm-Up1Exercises
EXAMPLE
Use lengths to find a geometric probability
Find the probability that a point chosen at random
on PQ is on RS .
SOLUTION
P(Point is on RS) =
Length of RS
Length of PQ
6
3
4 – ( – 2)
= , 0.6, or 60%.
=
=
10 5
5 – ( – 5)
Warm-Up2Exercises
EXAMPLE
Use a segment to model a real-world probability
MONORAIL
A monorail runs every 12 minutes. The ride from the
station near your home to the station near your work
takes 9 minutes. One morning, you arrive at the station
near your home at 8:46. You want to get to the station
near your work by 8:58. What is the probability you will
get there by 8:58?
SOLUTION
STEP 1
Find: the longest you can wait for the monorail and still
get to the station near your work by 8:58. The ride takes
9 minutes, so you need to catch the monorail no later
than 9 minutes before 8:58, or by 8:49. The longest you
can wait is 3 minutes (8:49 – 8:46 = 3 min).
Warm-Up2Exercises
EXAMPLE
Use a segment to model a real-world probability
STEP 2
Model the situation. The monorail runs every 12
minutes, so it will arrive in 12 minutes or less. You
need it to arrive within 3 minutes.
The monorail needs to arrive within the first 3 minutes.
STEP 3
Find: the probability.
P(you get to the station by 8:58)
Favorable waiting time
3
1
= Maximum waiting time =
=
ANSWER
12 4
The probability that you arrive by 8:58. is 1 or 25%.
4
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
Find the probability that a point chosen at random on PQ is
on the given segment. Express your answer as a fraction, a
decimal, and a percent.
1. RT
ANSWER
1
, 0.1, 10%
10
2.
TS
3.
PT
4.
RQ
ANSWER
ANSWER
ANSWER
1 , 0.5, 50%
2
2 , 0.4, 40%
5
7
, 0.7, 70%
10
Warm-Up3Exercises
EXAMPLE
Use areas to find a geometric probability
ARCHERY
The diameter of the target
shown at the right is 80
centimeters. The diameter of
the red circle on the target is 16
centimeters. An arrow is shot
and hits the target. If the arrow
is equally likely to land on any
point on the target, what is the
probability that it lands in the
red circle?
Warm-Up3Exercises
EXAMPLE
Use areas to find a geometric probability
Find the ratio of the area of the red circle
to the area of the target.
Area of red circle
P(arrow lands in red region) =
Area of target
SOLUTION
ANSWER
=
(82)
(402)
=
64
1600
1
=
25
The probability that the arrow lands in the red region is
1 ,
or 4%.
25
Warm-Up4Exercises
EXAMPLE
Estimate area on a grid to find a probability
SCALE DRAWING
Your dog dropped a ball in a
park. A scale drawing of the
park is shown.
If the ball is equally likely to be
anywhere in the park, estimate
the probability that it is in the
field.
SOLUTION
STEP 1
Find the area of the field. The shape is a rectangle,
so the area is bh = 10 3 = 30 square units.
Warm-Up4Exercises
EXAMPLE
Estimate area on a grid to find a probability
STEP 2
Find the total area of the park.
Count the squares that are fully
covered. There are 30 squares in
the field and 22 in the woods.
So, there are 52 full squares.
Make groups of partially covered squares so the combined
area of each group is about 1 square unit.
The total area of the partial squares is about 6 or 7 square
units.
So, use 52 + 6.5 = 58.5 square units for the total area.
Warm-Up4Exercises
EXAMPLE
Estimate area on a grid to find a probability
STEP 3 Write a ratio of the areas to find the probability.
Area of field
P(ball in field) =
Total area of park
30
58.5
300
= 585
20
= 39
ANSWER
The probability that the
ball is in the field is about
20
39 , or 51.3%.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
6. In the target in Example 3, each ring
is 8 centimeters wide. Find the
probability that an arrow lands in a
black region.
ANSWER
14 = 56%
25
7. In Example 4, estimate the
probability that the ball is in
the woods.
ANSWER
19 or about 48.7%.
39
Warm-Up
Exercises
Daily
Homework
Quiz
1.
Find the probability that a point chosen at random
on JN is on KN.
ANSWER
5 or 71.4%
7
2. Find the probability that a
randomly chosen point in the
figure will lie in the shaded region.
ANSWER
36.3%
Warm-Up
Exercises
Daily
Homework
Quiz
3.
A bird is equally likely to be anywhere in the
garden shown. What is the probability that the
bird will be in the vegetables?
ANSWER
about 24 or 47%
51