a We have two six-sided dice. We will list the possible sums that result from rolling these dice. To do so let's look at all outcomes of two dice in the figure in the example, and sum the numbers in these outcomes.
From the table we can see the possible sums that result from rolling two dice.
Possible Sums
2,3,4,5,6,7,8,9,10,11,12
b Now we will find the theoretical probability of rolling each sum. Therefore, we need to find the number of times that the sums occur by using the table in Part A. Also, the total number of outcomes is 36. From here we are ready to find the theoretical probabilities.
Sum
Frequency
Probability
2
1
1/36
3
2
2/36=1/18
4
3
3/36=1/12
5
4
4/36=1/9
6
5
5/36
7
6
6/36=1/6
8
5
5/36
9
4
4/36=1/9
10
3
3/36=1/12
11
2
2/36=1/18
12
1
1/36
c We will use a random number generator to simulate rolling two dice 50 times, then find the sum of the numbers on the dice. We want to compare the experimental probabilities of rolling each sum with the theoretical probabilities.
We will make an example table by using the above random number generator.
From here we will count the frequencies of each sum. Then we will find the experimental probabilities in which the number of trials is 50.
Sum
Frequency
Experimental Probabilities
2
2
2/50=1/25
3
3
3/50
4
4
4/50=2/25
5
7
7/50
6
4
4/50=2/25
7
9
9/50
8
10
10/50=1/5
9
5
5/50=1/10
10
3
3/50
11
1
1/50
12
2
2/50=1/25
To compare the probabilities we will use the theoretical probabilities that we found in Part B. Then, to make the comparison more apparent we will write the probabilities in decimal forms.
Sum
Experimental Probabilities
Theoretical Probabilities
2
1/25=0.04
1/36≈ 0.028
3
3/50=0.06
1/18≈ 0.06
4
2/25=0.08
1/12≈ 0.08
5
7/50=0.14
1/9≈ 0.11
6
2/25=0.04
5/36≈ 0.14
7
9/50=0.18
1/6≈ 0.17
8
1/5=0.1
5/36≈ 0.14
9
1/10=0.1
1/9≈ 0.11
10
3/50=0.06
1/12≈ 0.08
11
1/50=0.02
1/18≈ 0.06
12
1/25=0.04
1/36≈ 0.03
When looking at the probabilities for each sum, we can see that the probabilities are similar.
