CC-14
Guided Instruction
PURPOSE To show how individual and cumulative
probabilities can be displayed in a table or graph
PROCESS ​Students will
•use experimental and theoretical probabilities to
find individual probabilities.
•sum individual probabilities to find cumulative
probabilities.
Probability
Distributions
Common Core State Standards
MACC.912.S-IC.1.2 Decide if a specified model is
consistent with results from a given data-generating
process, e.g., using simulation.
MP 1
You can
distribu
A probability distribution is a function that gives the probability of each
outcome in a sample space. You can use a frequency table or a graph to show a
probability distribution.
5. Cop
cum
The theoretical probability of rolling each number on a standard number cube is the
same: 16. It is a uniform distribution, a probability distribution that is equal for each
event in the sample space. Here is a table and graph of its probability distribution.
DISCUSS ​Probability distributions are used to find
S
Event: Roll
1
2
3
4
5
6
Frequency
1
1
1
1
1
1
Probability
1
6
1
6
1
6
1
6
1
6
1
6
Probability
Rolling a Number Cube
probabilities of many different events using the
same sample space. Elicit that
•the sum of all possible probabilities in the same
sample space is equal to 1.
•cumulative values are found by adding the
included value and the values below it.
C
P
5
6
3
6
A
su
1
6
6. Rea
1
Activity 1
2
Now, suppose you roll two standard number cubes. You can show the probability
distribution for the sum of the numbers by making a frequency table and drawing
a graph.
If time permits, you may suggest that students
increase the number of trials so their data will be
closer to the theoretical probabilities. Students
could also combine data from their experiments.
3 4
Number
5
6
7. Cop
1.0
0.8
Probability
1
Roll a pair of standard number cubes 36 times. Record the sum for each roll.
Q Is this activity using experimental or theoretical
1. Copy the frequency table below. Use your data to complete your table.
probability? Explain. ​[Experimental; you are
actually rolling the number cubes and collecting
the data.]
Q Might your graph look different if you used
theoretical probabilities? Explain. ​[Yes;
experimental probability might show
probabilities of 0%. Theoretically, each of these
sums will occur.]
2
3
4
5
6
7
8
9
10
11
12
Frequency
■
■
■
■
■
■
■
■
■
■
■
Probability
■
■
■
■
■
■
■
■
■
■
■
Mathematical Practices CC-14 supports
students in making sense of problems,
Mathematical Practice 1.
52
Concept Byte
52
Cumulative Probability
6.It includes all the possible outcomes.
1.0
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Probability
1–2. Check students’ work.
3.
5
36
3
36
1
36
Activity 2
5.
2 to 4
2 to 6
2 to 8
2 to 10
2 to 12
1
6
5
12
13
18
11
12
1
Common Core
6
8
10
12
0.4
0.2
2
4
6
8. a. I
t
b. I
E
9. Use
{red
Sum
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...
8.a.No; there are six ways of rolling
a sum of
7. The probability of
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rolling a sum of 7 four times in a row
8 10 12
1
1
1
1
1
is 6 6 6 6 = 1296 . Although
this is unlikely, rolling a sum of 7 four
times in a row out of six different ways
is possible.
b.Yes; there is only one way to roll a sum
of 2. The probability of rolling a sum
( )( )
4.a–b.Check students’ work.
Cumulative
Probability
4
( )( )
1
1
of 2 four times in a row is 36 36 1
1
1
36
= 1,679,616
.
36
Sum
Sum
2
()()()()
0.6
Sum
1 2 3 4 5 6 7 8 9 10 11 12
52 0.8
0
Exerc
10
36
6
36
2
36
Probability Distributions
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Common
Core
Activity 1
Probability
4. a. Reasoning Compare the graphs. Do you think the number
cubes you rolled are fair? Explain.
b. Explain why there are differences, if any, between the
theoretical model and the experimental model.
P (sum 2) + P (sum 3) + P (sum 4).]
0.4
0
Sum of Two Number Cubes
3. Make a graph of the probability distribution for the sums of two
number cubes rolled 36 times, based on the theoretical
probabilities of each sum.
probability that the sum is less than 5? ​[Add
0.6
0.2
Event: Sum
2. Copy and complete the graph at the right using your data.
Q How can you use your table to find the
Answers
When yo
number
Cumula
than or
When you can assign numerical values to events, the cumulative frequency is the
number of times events with values that are less than or equal to a given value occurs.
Cumulative probability is the probability of events occuring with values that are less
than or equal to a given value.
Activity 2
Q What is the difference between a cumulative
You can use the data you collected in Activity 1 to construct a cumulative probability
distribution.
frequency distribution and a cumulative probability
distribution? [The cumulative frequency
2
distribution gives the number of times outcomes
occur at or below a certain value. The cumulative
probability distribution is the ratio of the
cumulative frequency over the sample space, or
the probability an outcome occurs at or below a
certain point.]
5. Copy and complete the table below. Add the probabilities within each range to find the
cumulative probalities.
Sum
Cumulative
Probability
2 to 4
2 to 6
■
■
2 to 8 2 to 10 2 to 12
■
Add the probabilities for the
sums of 2, 3, and 4.
■
Q Why does each cumulative probability include the
■
probabilities of the previous sums? [The previous
sums are less than the following sums and so are
included in “less than or equal to.”]
Add the probabilities for sums of
5 and 6 to the previous total.
Q Are there any possible outcomes besides the
integers from 2 to 12? Explain. How does this
help you answer Exercise 6? [No; the least sum
6. Reasoning Explain why the cumulative probability in the last interval is 1.
7. Copy the graph below and complete it using the cumulative probabilites you computed.
is 1 + 1 = 2 and the greatest is 6 + 6 = 12.
Since all possible outcomes are included,
the cumulative probability of all the sums
must be 1.]
Cumulative Probability Distribution
1.0
Probability
0.8
0.6
0.4
0.2
0
2
4
6
Sum
8
10
Extension
Cumulative probability tables and graphs can also
be used to find “greater than” probabilities. Use
complements to calculate these values.
12
Exercises
Exercises
Q For Exercise 9, what do you need to make the
table? [the event (color), the frequency of each
color, and the probability of each color]
Q How will you label your graph for Exercise 10? [The
y-axis will be Probability and the x-axis will be
Number of Days. The y-axis will go from 0 to 1
and the x-axis from 28 to 31.]
8. a. If you roll a pair of number cubes to model a situation and observe a sum of 7 four
times in a row, would you question the model? Explain.
b. If you observed a sum of 2 four times in a row, would you question the model?
Explain.
9. Use a table and a graph to show the probability distribution for the spinner
{red, green, blue, yellow}.
Concept Byte
53
Probability Distributions
CC-14 Probability/120/PE01457/TRANSITION_KITS/NA/ANCILLARY/2015/XXXXXXXXXX/Layout/Interior_Files/S
Distributions
...
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9.
3
Red
Green
Blue
Frequency
1
3
2
1
Probability
1
7
3
7
2
7
1
7
7
Yellow
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Probability
Event: Color
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2
7
1
7
0
red
green
blue
yellow
Color
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