Five-Minute Check (over Lesson 13–3)
CCSS
Then/Now
New Vocabulary
Key Concept: Designing a Simulation
Example 1: Design a Simulation by Using a Geometric Model
Example 2: Design a Simulation by Using Random Numbers
Example 3: Conduct and Summarize Data from a Simulation
Key Concept: Calculating Expected Value
Example 4: Calculate Expected Value
Over Lesson 13–3
A.
B.
C.
D.
Over Lesson 13–3
A.
B.
C.
D.
Over Lesson 13–3
A.
B.
C.
D.
Over Lesson 13–3
Camla knows the bus she needs comes every
hour and a half. What is the probability that
Camla waits 15 minutes or less for the bus?
A.
B.
C.
D.
Over Lesson 13–3
Find the probability that a point
chosen at random from inside the
circle lies in the shaded region.
A. 45.8%
B. 51.6%
C. 61.1%
D. 72.8%
Over Lesson 13–3
A spinner has 5 equal sections that are red, blue,
red, blue, red. What is the probability of the
pointer landing on blue?
A. 20%
B. 40%
C. 60%
D. 80%
Content Standards
G.MG.3 Apply geometric methods to solve problems
(e.g., designing an object or structure to satisfy
physical constraints or minimize cost; working with
typographic grid systems based on ratios).
S.MD.6 (+) Use probabilities to make fair decisions
(e.g., drawing by lots, using a random number
generator).
Mathematical Practices
1 Make sense of problems and persevere in solving
them.
4 Model with mathematics.
You found probabilities by using geometric
measures.
• Design simulations to estimate probabilities.
• Summarize data from simulations.
• probability model
• simulation
• random variable
• expected value
• Law of Large Numbers
Design a Simulation by Using a Geometric
Model
BASEBALL Maria got a hit 40% of the time she
was at bat last season. Design a simulation that
can be used to estimate the probability that she
will get a hit at her next at bat this season.
Step 1
Step 2
Possible Outcomes
Theoretical Probability
Maria gets a hit
Maria gets out
40%
(100 – 40)% or 60%
Our simulation will consist of 40 trials.
Design a Simulation by Using a Geometric
Model
Step 3
One device that could be used is a spinner
divided into two sectors, one containing 40%
of the spinner’s area and the other 60%. To
create such a spinner, find the measure of the
central angle of each sector.
Get a hit: 40% of 360° = 144°
Get out: 60% of 360° = 216°
Step 4
A trial, one spin of the spinner,
will represent one at-bat. A
successful trial will be a hit
and a failed trial will be getting
out. The simulation will consist
of 40 trials.
GAME SHOWS You are on a game show in which
you pull one key out of a bag that contains four
keys. If the key starts the car in front of you, then
you have won it. Which of the following geometric
spinners accurately reflects your chances of
winning?
A.
B.
C.
D.
Design a Simulation by Using Random
Numbers
PIZZA A survey of Longmeadow High School
students found that 30% preferred cheese pizza,
30% preferred pepperoni, 20% preferred peppers
and onions, and 20% preferred sausage. Design
a simulation that can be used to estimate the
probability that a Longmeadow High School
student prefers each of these choices.
Step 1
Possible Outcomes
Cheese
Pepperoni
Peppers and onions
Sausage
Theoretical Probability
30%
30%
20%
20%
Design a Simulation by Using Random
Numbers
Step 2
We assume a student’s preferred pizza type
will fall into one of these four categories.
Step 3
Use the random number generator on your
calculator. Assign the ten integers 0–9 to
accurately represent the probability data.
The actual numbers chosen to represent the
outcomes do not matter.
Outcome
Cheese
Pepperoni
Peppers and onions
Sausage
Represented by
0, 1, 2
3, 4, 5
6, 7
8, 9
Design a Simulation by Using Random
Numbers
Step 4
A trial will consist of selecting a student at
random and recording his or her pizza
preference. The simulation will consist of
20 trials.
PETS A survey of Mountain Ridge High School
students found that 20% wanted fish as pets, 40%
wanted a dog, 30% wanted a cat, and 10% wanted a
turtle. Which assignment of the ten integers 0–9
accurately reflects this data for a random number
simulation?
A. Fish: 0, 1, 2
Dog: 3, 4, 5, 6
Cat: 7, 8
Turtle: 9
B. Fish: 0, 1
Dog: 2, 3, 4, 5
Cat: 6, 7
Turtle: 8, 9
C. Fish: 0, 1
Dog: 2, 3, 4, 5, 6
Cat: 7, 8
Turtle: 9
D. Fish: 0, 1
Dog: 2, 3, 4, 5
Cat: 6, 7, 8
Turtle: 9
Conduct and Summarize Data from a
Simulation
BASEBALL Refer to the simulation in Example 1.
Conduct the simulation and report the results,
using the appropriate numerical and graphical
summaries.
Maria got a hit 40% of the time she was at bat last
season.
Make a frequency table and record the results after
spinning the spinner 40 times.
Conduct and Summarize Data from a
Simulation
Based on the simulation data, calculate the
probability that Maria will get a hit at her next at-bat.
This is an experimental
0.35
probability.
The probability that Maria makes her next hit is 0.35
or 35%. Notice that this is close to the theoretical
probability, 40%. So, the experimental probability of
her getting out at the next at-bat is 1 – 0.35 or 65%.
Make a bar graph of these results.
Conduct and Summarize Data from a
Simulation
Which one of these statements is not true about
conducting a simulation to find probability?
A. The experimental probability and the
theoretical probability do not have to be
equal probabilities.
B. The more trials executed, generally the
closer the experimental probability will
be to the theoretical probability.
C. Previous trials have an effect on the
possible outcomes of future trials.
D. Theoretical probability can be calculated
without carrying out experimental trials.
Calculate Expected Value
ARCHERY Suppose that an arrow
is shot at a target. The radius of
the center circle is 3 inches, and
each successive circle has a
radius 5 inches greater than that
of the previous circle. The point
value for each region is shown.
A. Let the random variable Y represent the point
value assigned to a region on the target. Calculate
the expected value E(Y) for each shot of the arrow.
First, calculate the geometric probability of landing in
each region.
Calculate Expected Value
Calculate Expected Value
Answer: The expected value of each throw is
about 4.21.
Calculate Expected Value
ARCHERY Suppose that an arrow
is shot at a target. The radius of
the center circle is 3 inches, and
each successive circle has a
radius 5 inches greater than that
of the previous circle. The point
value for each region is shown.
B. Design a simulation to estimate the average
value or the average of the results of your
simulation of shooting this game. How does this
value compare with the expected value you found
in part a?
Calculate Expected Value
Assign the integers 0–324 to accurately represent the
probability data.
Region 10 = integers 1–9
Region 8 = integers 10–64
Region 5 = integers 65–169
Region 2 = integers 170–324
Use a graphing calculator to generate 50 trials of
random integers from 1 to 324. Record the results in a
frequency table. Then calculate the average value of
the outcomes.
Calculate Expected Value
Answer: The average value 5.62 is greater than
the expected value 4.21.
A. In a similar situation to Example 4a, if the
following are the geometric probabilities of a target
with 3 regions, what is the expected value?
Assume each region is worth the value it is named.
A. 5.4
B. 6.3
C. 7.9
D. 8.7
B. If the chart is populated
by data from a simulation
and each region is worth the
value it is named, calculate
the average value from these
50 trials.
A. 5.9
B. 6.6
C. 7.1
D. 8.3