Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Probability and Statistics
Activity:
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TEKS:
(6.9) Probability and statistics. The student uses experimental and
theoretical probability to make predictions.
The student is expected to:
(A) construct sample spaces using lists and tree diagrams; and
(B) find the probabilities of a simple event and its complement and
describe the relationship between the two.
(6.13) Underlying processes and mathematical tools. The student
uses logical reasoning to make conjectures and verify conclusions.
The student is expected to:
(A) make conjectures from patterns or sets of examples and nonexamples; and
Overview:
Students will perform simple experiments and investigate the relationship
between the experimental probabilities of a simple event and its
complement. Group data will be collected and students will compare the
group data with the theoretical probability of the event.
Materials:
Number cubes/dice or a six section spinner (enough for every student to
have one)
Complements Come Easily Handout
Group Data Collection Sheet
Grouping:
Individual and small groups (3 or 4)
Time:
One 45 minute class
Lesson:
1.
Procedures
To set up this lesson, ask students to define
the following:
Experiment
Event
Sample space
Notes
Students can define by example,
such as an experiment is rolling a
number cube, drawing a card from
a standard deck of 52 cards, or
spinning a spinner. An
experiment is an action where the
results can be recorded.
An event is something we are
looking for, such as a roll of 5, or
the ace of spades, or red on a
spinner.
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Mathematics TEKS Refinement 2006 – 6-8
Procedures
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Notes
A sample space is a list of all
possible outcomes. A sample
space can also be shown in a tree
diagram. (A tree diagram is most
helpful when the outcomes are not
equally likely to occur.)
2.
Distribute the Complements Come Easily!
handout.
Make sure students understand
that they need to keep a record of
each roll. Each student will report
Explain that each student will collect data,
to the small group the number of
then combine that data with the data from the 1’s, 2’s, 3’s, etc.
rest of the group, and finally, all group data
will be collected to make up class data.
It is important that students predict
first what they think the probability
of rolling a 3 is.
3.
Walk around the room monitoring the
progress of the students. As each student
finishes, have him/her report data to the
group recorder. When a group has finished,
have the group recorder put the data on a
transparency of the Group Data Collection
Sheet.
4.
When all group data has been collected,
have each student record three different
probabilities.
1. the probability of rolling a three using
his/her own personal data.
2. the probability of rolling a three using their
group data.
3. the probability of rolling a three using the
class data.
If is often helpful to have students
change these probabilities to
decimals so that the values can be
compared easily.
5.
We have calculated the experimental
probability of rolling a three on a number
cube using three different groups of data.
What is the sample space for the
event “Roll a three”? {1, 2, 3, 4, 5,
6}
Actually we could compute the probability of
rolling a three mathematically. (Theoretical
probability)
Is it more likely to roll one number
than another? No (This
addresses the idea of equally
likely events.)
How many ways can we roll a
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Mathematics TEKS Refinement 2006 – 6-8
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Procedures
Notes
three? One
How many different numbers can
be rolled? Six
So the Theoretical Probability of
1
rolling a three is .
6
What is the decimal approximation
1
of ? 0.16666…
6
6.
Which of the probabilities that you computed
is closest to one-sixth?
The class data should be closest
in almost every situation. This is
the theory of large numbers. The
more trials of an experiment, the
closer to the theoretical probability
the experimental probability will
be.
This is a great place to use
technology to extend the number
of rolls to 1000 or greater.
This is a screen shot from the
Probability simulator on the TI-83.
Number of rolls, 100.
Note that even with just 100 rolls
the bars are close to the same
height.
7.
Now have students return to their own data
and consider the event B, the roll is not a
three.
Again, have students find the probability of
Event B using the three different sets of data.
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What do you notice about the
probability of Event B?
Answers will vary.
Probability of event A + probability
of event B = 1.
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Procedures
(their own, their group’s, and the class data)
Tarleton State University
Notes
Subtract Probability of Event A
from 1 to get Probability of Event
B.
If students need more practice
with this, there is data for all six
possible rolls.
Other events that could be
explored include:
Event C: roll is an even number.
Complement: roll is not an even
number.
Event D: roll is a prime number.
Complement: roll is not a prime
number.
Event E: roll is a square number.
Complement: roll is not a square
number.
8.
Event A and Event B are complementary
events.
Have students add this term to
their mathematics vocabulary list.
9.
Ask students to brainstorm a variety of
experiments and identify events for those
experiments and complementary events.
Encourage students to use
spinners, playing cards, or colored
counters for experiments.
10.
Reflection:
Have students define each of the following
and give an example.
Experiment
Event
Complementary Event
Sample space
Experimental Probability
Theoretical Probability
Homework:
Design an experiment, and identify an event. Conduct the experiment
and compute both the experimental and theoretical probability for the
event and its complement. The experiment cannot be roll a standard
number cube.
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Mathematics TEKS Refinement 2006 – 6-8
Assessment:
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The reflection serves as an assessment for this lesson.
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Mathematics TEKS Refinement 2006 – 6-8
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Consider the experiment: Roll a number cube.
What are the possible outcomes? (sample space) _____________________________
If you roll the number cube 30 times, predict how many times you will roll a 3. ________
(Event A: roll of a number cube is 3)
Write your prediction as a fraction;
number of 3′ s predicted
______________
total number of rolls
Now conduct the experiment by rolling a number cube 30 times, making a record of
what is rolled each time.
Based on the results of your 30 trials, write the probability of rolling a 3;
number of 3′ s rolled
_________________
total number of rolls
Compare the experimental probability of rolling a three with your prediction above.
What do you notice?____________________________________________________
Now, combine your data with the data from the other members of your group.
P(Event A group data) = ________
Compare the experimental probability of rolling a three with your prediction above.
What do you notice? ____________________________________________________
Record your group data on the Group Data Collection Sheet. The group will need to
report the number of 1’s, 2’s, etc. rolled and the total number of rolls.
P(Event A class data) = _____________
Now consider Event B: the roll is not a three.
Using your personal data, what is the experimental probability of Event B? __________
Using your group data, what is the experimental probability of Event B? _____________
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
How did you find this probability? ___________________________________________
______________________________________________________________________
Is there another way to find this probability? ________If so, explain.________________
______________________________________________________________________
Event A (roll is a 3)
Fraction
Decimal
Prediction
Event B (roll is not a 3)
Fraction
Decimal
XXXXXXXX
XXXXXXXX
Individual Data
Group Data
Class Data
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Group Data Collection Sheet
1
2
3
4
5
6
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
Group 7
Group 8
Group 9
Group 10
Total
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
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Possible answers
Consider the experiment: Roll a number cube.
What are the possible outcomes? (sample space) ____{1, 2, 3, 4, 5, 6}_____________
If you roll the number cube 30 times, predict how many times you will roll a 3. __5 or 6_
(Event A: roll of a number cube is 3)
Write your prediction as a fraction;
number of 3′ s predicted
5
_______ _______
total number of rolls
30
Now conduct the experiment by rolling a number cube 30 times, making a record of
what is rolled each time.
Roll
Tally
Frequency
1
2
2
8
3
6
4
4
5
6
6
4
Based on the results of your 30 trials, write the probability of rolling a 3;
number of 3′ s rolled
6
________
_________
total number of rolls
30
Compare the experimental probability of rolling a three with your prediction above.
What do you notice? ___answers will vary__________________________________
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Now, combine your data with the data from the other members of your group.
P(Event A group data) = ________
Compare the experimental probability of rolling a three with your prediction above. What
do you notice? ___answers will vary_________________________________________
Record your group data on the Group Data Collection Sheet. The group will need to
report the number of 1’s, 2’s, etc. rolled and the total number of rolls.
P(Event A class data) = _____
114
________
630
Now consider Event B= the roll is not a three.
Using your personal data, what is the experimental probability of Event B? ____
24
____
30
Using your group data what is the experimental probability of Event B? _____________
How did you find this probability? ___________________________________________
Is there another way to find this probability? ________If so, explain.________________
Prediction
Event A (roll is a 3)
Fraction
Decimal
5
0.166…
30
Event B (roll is not a 3)
Fraction
Decimal
XXXXXXXX
XXXXXXXX
Individual Data
6
30
0.2
24
30
0.8
Group Data
20
90
0.222…
70
90
0.777…
Class Data
114
630
0.18095
516
630
0.81905
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Group Data Collection Sheet
Sample Data
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
Group 7
Group 8
Group 9
Group 10
Total
1
2
3
4
5
6
14
20
20
12
13
11
19
12
12
15
12
20
9
14
24
16
16
11
12
16
13
14
20
15
13
15
15
14
19
14
15
15
17
16
21
6
16
16
13
14
15
16
98
108
114
101
116
93
Total Rolls 630
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Page 11