a Follow the given instructions and record the results using a table.
B
b Use a random number generator on a calculator.
C
c Use the results from the previous parts to fill the given table.
D
d Notice that the possible sums are integers between 2 and 12.
E
e Analyze the graphs from the previous part.
F
f Display the results from a random number generator using a bar graph.
G
g Can you see similarities between the two bar graphs?
H
h Think of an expected value as the most frequently occurring value.
A
aExample Solution:
Sum of Die Roll
7
5
3
11
3
11
8
7
6
4
9
8
5
7
5
4
8
6
7
5
B
bExample Solution:
Sum of Output from Random Number Generator
5
11
6
10
7
3
10
7
8
10
8
6
4
4
7
7
10
8
4
2
C
cExample Solution:
Trial
Sum of Die Roll
Sum of Output from Random Number Generator
1
7
5
2
5
11
3
3
6
4
11
10
5
3
7
6
11
3
7
8
10
8
7
7
9
6
8
10
4
10
11
9
8
12
8
6
13
5
4
14
7
4
15
5
7
16
4
7
17
8
10
18
6
8
19
7
4
20
5
2
D
d See solution.
E
eExample Solution: The bar graph has more data in its middle part as more outcomes are added.
F
fExample Solution:
G
gExample Solution: In both graphs most data are observed for the middle sums.
H
h See solution.
Practice makes perfect
a We are asked to roll two dice 20 times and record the sum of each roll.
Let's do this and record the obtained sums in a table.
Sum of Die Roll
7
5
3
11
3
11
8
7
6
4
9
8
5
7
5
4
8
6
7
5
b This time we are asked to use the random number generator on a calculator to generate 20 pairs of integers between 1 and 6. To do so, push the MATH button. Then scroll left to the PRB menu and choose the fifth option, randInt(.
After choosing this option, enter the minimum and maximum values of the set and the number of trials. Next, push ENTER. For our experiment, the minimum value is 1, the maximum is 6, and the number of trials is 2. Let's generate 20 such pairs.
c Now we will copy and complete the given table using our results from the previous parts.
Trial
Sum of Die Roll
Sum of Output from Random Number Generator
1
7
5
2
5
11
3
3
6
4
11
10
5
3
7
6
11
3
7
8
10
8
7
7
9
6
8
10
4
10
11
9
8
12
8
6
13
5
4
14
7
4
15
5
7
16
4
7
17
8
10
18
6
8
19
7
4
20
5
2
d Since we want to graph the number of times each possible sum occurred in the first 5, 10, and 20 outcomes, let's list all of the possible sums.
2,3,4,5,6,7,8,9,10,11,12
Therefore, the possible sums are integers between 2 — when we roll 1 on both dice — and 12 — when we roll 6 on both dice. Let's start with displaying the first 5 outcomes using a bar graph. To do this we can use the table we created in Part A.
Next, let's repeat the process for the first 10 outcomes.
Finally, let's draw a bar graph for all 20 outcomes.
e Based on the graphs we created in the previous part we can see that as more outcomes are added the bar graph has more data in its middle part.
f This time let's create a bar graph for outcomes from a random number generator. Again, on the vertical axis we will have integers from 2 to 12.
g Looking at the graphs of the die trial and the random number trial, we can see that in both graphs the most data are observed for the middle sums.
An expected value is the average value of a random variable that one expects after repeating an experiment or simulation a theoretically infinite number of times.
This means that based on our graphs, we could suspect that 7 will be an expected value because it it the most frequently occurring value for both an experiment and a simulation. However, since we did only 20 trials — which is far from an infinite number of times — we should calculate the expected value to make sure.
