Download Using Tree Diagrams to Calculate Probabilities

Using Tree Diagrams to
Calculate Probabilities
Objective To use tree diagrams to help calculate probabilities.
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Common
Core State
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Ongoing Learning & Practice
Key Concepts and Skills
Writing and Graphing Inequalities
• Use a unit percent to calculate the percent
of a number. [Number and Numeration Goal 2]
Math Journal 2, pp. 261A and 261B
Students write and graph inequalities
to represent descriptions of various
objects and situations in space.
• Add and subtract fractions with like and
unlike denominators. [Operations and Computation Goal 3]
• Use an algorithm to multiply a fraction by a
fraction and a whole number by a fraction. [Operations and Computation Goal 4]
• Determine the number of possible
outcomes for situations using diagrams or
counting strategies. [Data and Chance Goal 3]
• Calculate the probabilities of outcomes
using a tree diagram. [Data and Chance Goal 3]
Math Boxes 7 5
Math Journal 2, p. 262
Students practice and maintain skills
through Math Box problems.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Adding Fractions
Students review methods for adding
fractions with unlike denominators.
EXTRA PRACTICE
Experimenting with Coin Flips
Study Link 7 5
Math Masters, p. 232
per partnership: 1 coin (optional)
Students use a tree diagram to predict
outcomes in a coin-flipping experiment.
Math Masters, p. 231
Students practice and maintain skills
through Study Link activities.
5-Minute Math
Key Activities
Students use tree diagrams to find expected
outcomes and calculate the probabilities of
those outcomes.
EXTRA PRACTICE
5-Minute Math™, pp. 130 and 209
Students apply the Multiplication Counting
Principle to solve problems.
Ongoing Assessment:
Informing Instruction See page 648.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 261. [Data and Chance Goal 3]
Key Vocabulary
Multiplication Counting Principle
Materials
Math Journal 2, pp. 260 and 261
Study Link 74
transparency of Math Masters, p. 226
(optional)
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 172, 173
Lesson 7 5
645
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP6
Content Standards
Getting Started
Mental Math
and Reflexes
Students calculate the number of possible
outcomes. Suggestions:
rolling a die and tossing a coin
6 ∗ 2, or 12
choosing a dinner from 5 main
courses, 3 vegetables, 2 salads,
and 4 beverages
5 ∗ 3 ∗ 2 ∗ 4, or 120
selecting one hour of the day
and one minute of the hour
24 ∗ 60, or 1,440
6.EE.8
Math Message
Tim is late to school about 1 school day out of 10. At this rate, about how
many school days out of 30 would you expect Tim to be late? About how many
school days out of 200?
Study Link 7 4 Follow-Up
Review the answers as necessary. Briefly discuss how the lists and tree
diagram show that there are 12 possible breakfast bags. Instead of counting
all the possible choices, students can use the Multiplication Counting Principle
(also known as the Fundamental Counting Principle) as follows:
Number of
Beverage Choices
Number of
Bagel Choices
Number of
Fruit Choices
Total Number
of Choices
2
2
3
2 ∗ 2 ∗ 3 = 12
1 Teaching the Lesson
NOTE The Mental Math and
Reflexes problems deal with
compound events. For additional
practice with compound events,
see www.everydaymathonline.com.
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
The probability that Tim will be late on any particular school day
1 = 0.1 = 10%. The expected number of late arrivals depends
is _
10
on the number of days being considered.
1 of
Out of 30 school days, Tim is expected to be late _
30 days, or 3 days.
10
1 of
Out of 200 school days, Tim is expected to be late _
10
200 days, or 20 days.
▶ Revisiting Probability
WHOLE-CLASS
DISCUSSION
Tree Diagrams
(Math Journal 2, p. 260; Math Masters, p. 226)
The mazes shown in Problems 1 and 2 on journal page 260 are the
same as those from journal pages 257 and 258 of Lesson 7-4.
In Lesson 7-4, students were told the number of people who
entered a maze and then they were asked how many could be
expected to enter Rooms A and B. In this lesson, students
calculate the probability that a person walking through the maze
enters Room A and the probability that a person enters Room B.
646
Unit 7
Probability and Discrete Mathematics
Teaching Master
Work through Problem 1 with the class. If you have a transparency
of Math Masters, page 226, use the top diagram. Go through each
step carefully, making sure students understand the procedure.
Name
Date
LESSON
Time
Tree Diagrams
7 4
䉬
Step 1: Calculate the probability of reaching each of the
endpoints.
Because the tree diagram divides into four branches at the
first intersection, the probability of following any one of these
1 . Write _
1 next to each branch.
branches is 1 out of 4, or _
4
4
B
For each of the second set of intersections, the probability of
1 . Write _
1
following any one of the branches is 1 out of 2, or _
2
2
next to each branch.
B
Calculate the probabilities of reaching each of the endpoints of
1 of the time and
the tree diagram. The first branch is taken _
4
1
1
1 ∗_
1
_
_
each of the second branches taken 2 of 4 of the time, or _
2
4
1 ∗_
1 =_
1 , so the probability of reaching any one
of the time. _
2
4
8
1 . Record this probability at each endpoint.
of the endpoints is _
8
1
4
1
2
1
2
1
2
1
4
1
2
1
8
1
8
1
8
1
8
1
4
A
B
A
B
A
A
A
A
A
B
B
Math Masters, p. 226
1
4
1
2
1
2
1
2
1
2
1
8
1
8
1
8
1
8
Students should notice that, to calculate the probability of any of
the endpoints in the diagram, they multiply the probabilities of
1 ∗_
1 , or _
1.
the branches that lead to that endpoint; in this case, _
2
4
8
Step 2: Calculate the probabilities of entering each room.
Use the tree diagram to calculate the probability of entering
Room A and the probability of entering Room B.
There are five different endpoints in Room A. Add the
probabilities of reaching these endpoints. Since
5 , the probability of
1 +_
1 +_
1 +_
1 +_
1 =_
_
8
8
8
8
8
8
5.
entering Room A is _
8
There are three different endpoints in Room B. Add the
3,
1 +_
1 +_
1 =_
probabilities of reaching these endpoints. Since _
8
8
8
8
3
_
the probability of entering Room B is 8 .
Student Page
Date
Suppose 80 people walk through the maze. The probability of
5 . The number of people who can be
entering Room A is _
8
5 of 80 people, or 50 people.
expected to enter Room A is _
8
3
_
Similarly, 30 people ( 8 of 80 people) can be expected to enter
Room B.
Probability Tree Diagrams
75
䉬
Complete the tree diagram for each maze.
154 155
Write a fraction next to each branch to show the probability of selecting that branch.
Then calculate the probability of reaching each endpoint. Record your answers in the
blank spaces beneath the endpoints.
1.
5
ᎏᎏ
8
3
ᎏᎏ
8
What is the probability of entering Room A?
What is the probability of entering Room B?
1
4
From Step 2, we know the probability that a person will enter
Room A and the probability that a person will enter Room B. If a
large number of people enter the maze, the number expected to
enter each room can be found.
Example:
Time
LESSON
1
2
1
4
1
4
1 1
2 2
1 1
8 8
2.
1
4
1 1
2 2
1 1
8 8
Enter
1 1
2 2
1 1
8 8
1
2
Room A
1 1
8 8
Room B
5
ᎏᎏ
12
7
ᎏᎏ
12
What is the probability of entering Room A?
What is the probability of entering Room B?
1
3 1
3
1
6
1
6
1
2
1
2
1
3
1
2
1
6
1
4
Enter
1
2
Room A
Room B
1
4
Math Journal 2, p. 260
Lesson 7 5
647
Student Page
Date
Time
LESSON
Probability Tree Diagrams
75
䉬
3.
Adjusting the Activity
continued
Josh has 3 clean shirts (red, blue, and green) and 2 clean pairs of pants (tan and
black). He randomly selects one shirt. Then he randomly selects a pair of pants.
夹
a.
1
3
Shirts:
Refer students to journal page 257, where they worked with numbers on
the tree diagram. Relate the numbers at each step to the fraction of people
walking a given path. Remind students that the whole changes as each new
1
intersection is reached. In the beginning, the whole is 80 people and _
4 of 80
people, or 20 people, walk down each path. At the next intersection, the whole
is 20 people. One-half of 20 people, or 10 people, continue down each path
from there.
1
3
1
3
red
1
2
blue
1
2
1
2
Pants: tan black
1 1
6 6
b.
155 156
Complete the tree diagram by writing a fraction next to each branch to show the probability
of selecting that branch. Then calculate the probability of selecting each combination.
green
1
2
1
2
1
2
tan black
tan black
1 1
6 6
1 1
6 6
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
List all possible shirt-pants combinations. One has been done for you.
red-tan red-black, blue-tan, blue-black, green-tan, green-black
6 combinations
c.
How many different shirt-pants combinations are there?
d.
Do all the shirt-pants combinations have the same chance of being selected?
e.
What is the probability that Josh will select
1
ᎏᎏ
3
1
ᎏᎏ
2
the blue shirt?
the tan pants?
the blue shirt and the tan pants?
Yes
Ongoing Assessment: Informing Instruction
1
ᎏᎏ
6
2
ᎏᎏ
3
a shirt that is not red?
Watch for students who think that the correct strategy is to divide the original
total by the number of exits. After working through Problem 2, students should
see that this strategy is faulty.
1
ᎏᎏ
3
the black pants and a shirt that is not red?
Try This
Suppose Josh has 4 clean shirts and 3 clean pairs of pants. Explain how to calculate
the number of different shirt-pants combinations without drawing a tree diagram.
Multiply the number of clean shirts (4) by the number of
clean pants (3); 4 ⴱ 3 ⫽ 12
Ask students to work Problem 2 independently. Bring the class
together to discuss the answers. You might want to use the
diagram at the bottom of the transparency of Math Masters,
page 226 to discuss the problem. The probability of entering
7.
1 +_
1 +_
1 , or _
Room B is _
6
6
4
12
Math Journal 2, p. 261
1
2
1
3
1
3
1
2
1
3
1
1
1
6
6
6
Room B Room B Room A
75
Going to Space
Star masses are measured in a unit called a solar mass, which is equal to the mass of
the Sun. One solar mass is equal to 2 ∗ 1033 grams.
1.
Low-mass stars use hydrogen fuel so slowly that they may shine for over 100 billion years.
A low-mass star has a mass of at least 0.1 solar mass but less than 0.5 solar mass.
a.
b.
▶ Calculating Probabilities
List several possible masses that a low-mass star could have.
Sample answers: 0.1 solar mass; 0.45 solar mass;
0.32 solar mass
with Tree Diagrams
Describe in words the set of all possible masses of a low-mass star.
Sample answer: All masses between 0.1 and 0.5
solar mass, including 0.1
c.
2.
(Math Journal 2, p. 261)
Use an inequality or two inequalities to describe the set of possible masses.
Sample answer: m < 0.5 and m ≥ 0.1
d.
Graph the set of all possible masses on the number line below.
e.
Do the values represented on the graph make sense in the situation? Explain your answer.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
High-mass stars have short lives and sometimes become black holes. When a high-mass star
dies, an explosion occurs, leaving behind a stellar core. If the stellar core has a mass of at
least 3 solar masses, the star becomes a black hole.
b.
List several possible masses of a stellar core that will become a black hole.
Sample answers: 3 solar masses; 5.5 solar masses;
3.1 solar masses
Describe the set of possible masses in words.
Sample answer: All masses that are 3 solar masses
or greater
Math Journal 2, p. 261A
261A_261B_EMCS_S_G6_MJ2_U07_576442.indd 261A
648
Unit 7
INDEPENDENT
ACTIVITY
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VING
VI
VIN
IN
IN
NG
G
Read the introduction to the problem as a class, so students know
what to do. The tree diagram has already been drawn and labeled.
Students should quickly notice that the probability of taking any
1 ∗_
1 , or _
1.
path in Problem 3 is _
3
2
6
1.0
Sample answer: Yes. A mass can be any positive
number, and the graph does not show any negative
masses.
a.
1
1
4
4
Room A Room B
Ask: If 60 people walk through the maze, about how many would
5 of 60, or 25 people About how many would enter
enter Room A? _
12
7 of 60, or 35 people
Room B? _
12
Time
LESSON
1
2
5.
1 +_
1 , or _
The probability of entering Room A is _
6
4
12
Student Page
Date
1
2
3/4/11 10:22 AM
Probability and Discrete Mathematics
Student Page
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 261
Date
Time
LESSON
Going to Space
75
2. c.
Use journal page 261 to assess students’ abilities to determine expected
outcomes and to use a tree diagram to calculate probabilities of chance events.
Students are making adequate progress if they are able to complete the tree
diagram and list all the possible shirt-and-pants combinations (Problems 3a–3d).
Some students may be able to calculate all probabilities in Problem 3e.
continued
c≥3
Use an inequality to describe the set of possible masses.
d.
Graph the set of all possible masses on the number line below.
e.
Do the values represented on the graph make sense in the situation? Explain your answer.
0
1
2
3
4
5
6
7
8
9
10
Sample answer: Not all of the solutions make sense.
For example, it is probably not possible for a stellar
core to have 1,000 solar masses. But since I don’t
know the maximum mass, I can’t change the graph.
[Data and Chance Goal 3]
In the United States, people who pilot spacecrafts or work in space are called astronauts. In
Russia and other former republics of the Soviet Union, these people are called cosmonauts.
3.
Links to the Future
Cosmonauts began flying the Soyuz series of spacecraft in 1967. These vehicles can transport
up to 3 cosmonauts.
a.
List the possible numbers of cosmonauts that can go on a mission in a Soyuz spacecraft.
b.
Describe the possible numbers of cosmonauts in words.
c.
Write an inequality or inequalities to represent the possible numbers of cosmonauts.
d.
Graph the possible numbers of cosmonauts on the number line below.
e.
Do the values shown on the graph make sense in the situation? Explain your answer.
1, 2, or 3 cosmonauts
In this unit, students discover that they can determine the number of possible
outcomes by making an organized list, drawing a tree diagram, or multiplying.
Students will extend their knowledge of the Multiplication Counting Principle
when they study combinatorics in future probability and statistics courses.
Sample answer: At least 1 but no more than
3 cosmonauts
Sample answer: n ≥ 1 and n ≤ 3
0
1
2
3
4
5
6
7
8
9
10
Sample answer: Yes. I used dots on the whole
numbers because you can’t have part of a cosmonaut.
2 Ongoing Learning & Practice
▶ Writing and Graphing Inequalities
Math Journal 2, p. 261B
261A_261B_EMCS_S_G6_MJ2_U07_576442.indd 261B
3/4/11 10:22 AM
INDEPENDENT
ACTIVITY
(Math Journal 2, pp. 261A and 261B)
Review with students the steps for graphing inequalities. Ask:
How is graphing an inequality that represents a real-world
situation different from graphing an inequality without context?
If you are representing something in the real world, you need to
check whether all the solutions on your graph make sense for the
situation.
Have students complete journal pages 261A and 261B. They
will write and graph inequalities to represent various objects
and situations in space. When they have finished, discuss their
answers to Part e as a class.
Student Page
Date
Math Boxes
75
䉬
1.
▶ Math Boxes 7 5
Time
LESSON
Darnell has 3 jackets and 4 baseball hats.
Complete the tree diagram.
Hats
Braves
Jackets
Dodgers
Cubs
Black
INDEPENDENT
ACTIVITY
Marlins
a.
(Math Journal 2, p. 262)
Writing/Reasoning Have students write a response to the
following: Explain why the graph in Problem 2 does not
represent the set of counting numbers. Sample answer: The
graph is a solid arrow pointing to the right, which means that all
numbers between the counting numbers (1.5, 1.501, 1.5011, and
so on) are also included in the graph. A graph of only the counting
numbers would be filled-in dots on 1, 2, 3, ... .
D
Red
12
b.
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 7-7. The skills in Problems 4 and 5
preview Unit 8 content.
B
How many jacket-hat
combinations are possible?
Yes
2.
C
M
Do all the jacket-hat combinations have
the same chance of being selected?
Which set of numbers is represented by
the graph below? Choose the best answer.
⫺3
⫺2
B
D
C
M
White
⫺1
0
1
2
3.
3
positive real numbers
positive integers
155 156
Write each number in standard notation.
72,000,000,000
a.
72 billion
b.
42.78 million
c.
89.6 billion
89,600,000,000
d.
0.5 million
500,000
42,780,000
whole numbers
counting numbers
4
244
4.
Janella walks at a speed of 6.9 kilometers
per hour. At this rate, how far can she walk
b.
13.8 kilometers
in 20 minutes? 2.3 kilometers
c.
in 1 hour 40 minutes?
a.
5.
—
Express the length of JK to the length
—
of JL as a simplified fraction.
J
in 2 hours?
11.5 kilometers
6
K
110 111
JK
ᎏᎏ
JL
⫽
10
8
L
3
ᎏᎏ
5
179
Math Journal 2, p. 262
Lesson 7 5
649
Study Link Master
Name
Date
STUDY LINK
Time
▶ Study Link 7 5
A Random Draw and a Tree Diagram
75
䉬
154 155
I
T P
O
A
Box 1
(Math Masters, p. 231)
Home Connection Students solve a problem involving
random draws.
N E
Box 2
INDEPENDENT
ACTIVITY
Boxes 1, 2, and 3 contain letter tiles.
Box 3
Suppose you draw one letter from each box without looking. You lay the letters in a
row—the Box 1 letter first, the Box 2 letter second, and the Box 3 letter third.
Complete the tree diagram. Fill in the blanks to show the probability for each branch.
1.
1
ᎏᎏ
2
1
ᎏᎏ
2
Box 1 T
1
ᎏᎏ
3
Box 2 I
1
ᎏᎏ
2
1
ᎏᎏ
2
Box 3 N
1
ᎏᎏ
12
1
ᎏᎏ
3
1
ᎏᎏ O
3
1
ᎏᎏ
2
A
1
ᎏᎏ
2
P
1
ᎏᎏ
3
I
1
ᎏᎏ
2
1
ᎏᎏ
2
1
ᎏᎏ
2
1
ᎏᎏ
2
A
1
ᎏᎏ
2
1
ᎏᎏ
2
1
ᎏᎏ
2
1
ᎏᎏ
2
E
N
E
N
E
N
E
N
E
N
E
1
ᎏᎏ
12
1
ᎏᎏ
12
1
ᎏᎏ
12
1
ᎏᎏ
12
1
ᎏᎏ
12
1
ᎏᎏ
12
1
ᎏᎏ
12
1
ᎏᎏ
12
1
ᎏᎏ
12
1
ᎏᎏ
12
1
ᎏᎏ
12
2.
How many possible combinations of letter tiles are there?
3.
What is the probability of selecting:
1
ᎏᎏ
6
a. the letters P and I?
c.
3 Differentiation Options
1
ᎏᎏ
3
1
ᎏᎏ O
3
1
ᎏᎏ
3
the letter combinations TO or PO?
▶ Adding Fractions
12
b.
3
ᎏᎏ,
the letter I, O, or A? 3
d.
two consonants in a row?
36.5
5.
22.6
858.8 ⫼ 38 ⫽
0%
12.6
1,575 ⫼ 125 ⫽
6.
5–15 Min
or 100%
Practice
657 ⫼ 18 ⫽
4.
SMALL-GROUP
ACTIVITY
READINESS
Math Masters, p. 231
To provide students with more practice using fractions to calculate
probability, have them add fractions. Review the quick common
denominator (QCD) and least common denominator (LCD) methods
for adding fractions with unlike denominators by having students
find sums.
Suggestions:
5
2 +_
1 _
_
3
1
_
6
1
_
5
+
+
6
1
_
4
1
_
3
6
5
_
12
8
_
15
5
1 +_
1 _
_
2
8 8
5
2 +_
1 _
_
9
3 9
3 _
19
1 +_
2 +_
_
4
5
10 20
EXTRA PRACTICE
▶ Experimenting with Coin Flips
PARTNER
ACTIVITY
15–30 Min
(Math Masters, p. 232)
To provide students more practice with theoretical and
experimental probabilities, have them predict the results
of a coin-flipping experiment with the help of tree diagrams. They
analyze the results and answer questions about the experiment.
Teaching Master
Name
Date
LESSON
75
䉬
1.
Time
A Coin-Flipping Experiment
Suppose you flip a coin 3 times.
What is the probability that the coin will land
a. HEADS
c. HEADS
e.
3 times?
1
ᎏᎏ
8
b. HEADS
3
ᎏᎏ
8
1 time and TAILS 2 times?
1
ᎏᎏ
8
3 times?
d. TAILS
with the same side up all 3 times (that is, all
2 times and TAILS 1 time?
or all
HEADS
TAILS)?
3
ᎏᎏ
8
1
ᎏᎏ
4
Students can extend the activity by actually carrying out the
experiment, flipping a coin 300 times and recording the results.
They can then compare actual results with predicted results.
Make a tree diagram to help you solve the problems.
EXTRA PRACTICE
H1
T1
1
ᎏᎏ
2
1
ᎏᎏ
2
2nd flip: H2
3rd flip: H3
1
ᎏᎏ
2
1
ᎏᎏ
2
T3 H3
1
ᎏᎏ
8
2.
1
ᎏᎏ
2
1
ᎏᎏ
2
T2 H2
1
ᎏᎏ
2
1
ᎏᎏ
8
▶ 5-Minute Math
1
ᎏᎏ
2
1
ᎏᎏ
2
1st flip:
1
ᎏᎏ
8
1 1
ᎏᎏ ᎏᎏ
2 2
T3 H3
1
ᎏᎏ
8
1
ᎏᎏ
8
One trial of an experiment consists of flipping a coin
3 times. Suppose you perform 100 trials. For about how
many trials would you expect to get HHH or TTT?
What percent of the trials is that?
T2
1
ᎏᎏ
2
1
ᎏᎏ
2
T3 H3
1
ᎏᎏ
8
1
ᎏᎏ
8
1
ᎏᎏ
2
T3
1
ᎏᎏ
8
About 25 trials
25%
Math Masters, p. 232
650
Unit 7
Probability and Discrete Mathematics
SMALL-GROUP
ACTIVITY
5–15 Min
To offer more practice applying the Multiplication Counting
Principle (also known as the Fundamental Counting Principle),
see 5-Minute Math, pages 130 and 209.