a Start with thinking about the possible sums. Arrange them in a table. In what situations can each sum be obtained?
B
b Think about how we calculated the probability. Was an actual experiment conducted?
A
a
Sum
2
3
4
5
6
7
8
9
10
11
12
Frequency
1
2
3
4
5
6
5
4
3
2
1
Probability
1/36
1/18
1/12
1/9
5/36
1/6
5/36
1/9
1/12
1/18
1/36
B
b Theoretical. See solution.
Practice makes perfect
a We want to make a probability distribution for the sum of the faces after rolling two standard number cubes. To do so we first need to think about all the possible sums. Since the smallest number on each cube is 1, the least possible sum is 1+1=2. The greatest number on each cube is 6, so the greatest possible sum is 6+6=12. Every sum between 2 and 12 is also possible.
Sum
2
3
4
5
6
7
8
9
10
11
12
Frequency
Probability
We want to complete this table. First we need to find the frequencies for each sum. To do so, let's think about all possible ways to obtain each sum and count them. We will consider the order of the outcomes on the cubes.
Sum
Possibilities
Frequency
2
1+1
1
3
1+2, 2+1
2
4
1+3, 3+1, 2+2
3
5
1+4, 4+1, 2+3, 3+2
4
6
1+5, 5+1, 2+4, 4+2, 3+3
5
7
1+6, 6+1, 2+5, 5+2, 3+4, 4+3
6
8
2+6, 6+2, 3+5, 5+3, 4+4
5
9
3+6, 6+3, 4+5, 5+4
4
10
4+6, 6+4, 5+5
3
11
5+6, 6+5
2
12
6+6
1
Note that these frequencies are made assuming that all outcomes are equally probable, so when we conduct an experiment they may vary. However, we need this assumptions to make a correct probability distribution. Let's add the frequencies to our table!
Sum
2
3
4
5
6
7
8
9
10
11
12
Frequency
1
2
3
4
5
6
5
4
3
2
1
Probability
Finally, to calculate the probability we need to calculate the relative frequencies. To do so, we need to divide each frequency by the total of the frequencies. The total of the frequencies is the sum of the frequencies.
We can now calculate the probabilities. Let's do it!
Sum
2
3
4
5
6
7
8
9
10
11
12
Frequency
1
2
3
4
5
6
5
4
3
2
1
Probability
1/36
2/36=1/18
3/36=1/12
4/36=1/9
5/36
6/36=1/6
5/36
4/36=1/9
3/36=1/12
2/36=1/18
1/36
b We now want to determine if the probabilities are theoretical or based on experimental results. Consider our table.
Sum
2
3
4
5
6
7
8
9
10
11
12
Frequency
1
2
3
4
5
6
5
4
3
2
1
Probability
1/36
1/18
1/12
1/9
5/36
1/6
5/36
1/9
1/12
1/18
1/36
As we stated before, to find the frequencies we assumed that each outcome is equally probable. Therefore, the frequencies were not based on actual results of an experiment. Since we used them to calculate the probability, the probability is also not based on experimental results, so it is theoretical.
