Frequency of a Combination Equaling to Their Sum:

Rolling a pair of dice and analyzing the outcomes

 

 

 

 

 

Nahin Imtiaz

23^rd October 2019

 

 

 

Abstract:

We know that when we roll a dice, the probability of having an outcome is constant which is 1/6. But in our experiment, we worked with a pair of dice to check if the combinations equaling to their sum (i.e. 7 has combinations 3/4, 5/2, 6,1) have the same frequency of occurring. We rolled a pair of dice 100 times and recorded all possible combinations. Then we sorted them in categories corresponding to their sum. After that, we analyzed the sum that has most outcomes and found out that its combinations don’t have the same frequency of occurrence. This experiment showed that even though a number in a single dice has a constant probability of occurring when we consider two dice and a combination, it doesn’t remain the same.

        

 

 

Introduction:

Probability means the possibility of an event happening. It can be found by dividing the number of a specific event happening by the total number of events. For a single dice, possibility of each event happening has a fixed value which is 1/6. If we roll a dice, the number 6 will have the same probability of happening as the number of 5. In our experiment, we will work with pair of dice and see if the rate of occurring of a combination remains same for other combinations as well or not.

Materials and Methods:

The materials we need for this experiment are-

• A pair of dice
• A piece of paper and a pen

1. Two dices were taken in hand and rolled at the same time. For both dices, the numbers facing up were written down in the paper along with their sum value. This process was continued for 100 times. Since the process of rolling a pair of dice and recording the combinations is time-consuming, a different approach to the problem can be taken. The code from the Calculation can be used to get two random values ranging from 1 to 6, their summation value and combinations for each roll.
2. For each possible summation value of two dice ranging from 2 to 12, all combinations equaling to the summation value were counted and recorded.

3, From 2 to 12, the summation value that occurs the most, were selected. For the selected summation value, the occurrence of each combination equaling to that value was recorded and compared against each other to check if the frequency of each combination is same or not. To get the frequency, the equation of rate from Calculation was used.

Results:

Table 1 has two main columns named, Sum and Combination. Each row of Combination corresponds to the combinations that occurred during dice roll and the Sum corresponds to the summation of each of those combinations. For example, Sum 6 occurs from four combinations: (5,1),(5,1),(4,2),(2,4)

Table 1. Outcomes of 100 rolls

Figure 1 is a bar graph of the Sum of Two Dice vs Frequency. This bar graph shows the number of times a Sum value occurs. The graph tells us that 2,6 and 12 occur significantly less than 7 and 9.

Figure 1. Frequency of each summation value ranging from 2 to 12.

Figure 2. Frequency of each combination equaling to 7

Figure 2 has two axes: Combinations and Occurrence. Combination shows three possible combinations equaling to 7 and Occurrence shows the amount of time these combinations occur during dice roll.

Analysis:

According to William A. Ewbank and John L. Ginther , Mathematics Teaching in the Middle School showed in their article, Probability on a Budget (2002) that for a pair of trick dice, we will get combinations that will be used to determine who wins in a game. Combinations like (X,X),(Y,Y) will result in draws. In our experiment, we will check what’s the frequency of these events.

As Figure 1 shows, 7 occurs the most and occurs 21 times. 12 occurs only two times. We will work with 7 since it has a large number of occurrence and we know that a large number of sample space gives more accurate results. All combinations equaling to 7 are (3/4,5/2,1/6). Notice that we have chosen to omit combinations (4/3,2/5,6,1), since, in our experiment, order doesn’t matter, and the summation of 3/4 and the summation of 4/3 will be the same.

Figure 2 shows the frequency of each combination equaling to 7. Combination (3,4) has the highest rate of occurrence which is 10. Combination (1,6), whose rate of occurrence is 3, is the lowest among all of them. So, from the equation in Calculation we find that the rate of occurrence for combination (3,4), (5,2), (1,6) is 0.10, 0.08, 0.03 respectively. We find that combination (3,4) has the highest rate of occurrence while combination (1,6) has the lowest. Moreover, they are not equal. This proves that the combinations equaling to 7 (3/4,5/2,1/6) don’t occur with the same frequency. This will be true for other values of Sum ranging from 2 to 12 as well.

Conclusion:

         From our analysis, we can conclude that when a pair of dice is rolled, the combination we get from it doesn’t occur with the same frequency as the next dice roll. The reasoning behind it is that we are dealing with random events. Even though a pair of dice has fixed rate of occurrence, when we roll a pair of dice, they won’t have fixed rate of occurrence because each dice is independent and doesn’t depend on the other dice. For this reason, the rate of frequency of each combination cannot be the same.

Work Cited:

EWBANK, W., & GINTHER, J. (2002). PROBABILITY on a Budget. Mathematics Teaching in the Middle School, 7(5), 280-284. Retrieved from http://www.jstor.org.ccny-proxy1.libr.ccny.cuny.edu/stable/41181139

Appendix:

The data we get from our program will be similar to the data given below. Note that we are working with random function. So, the data will change each time we run the program.

10(5,5)  10(6,4)  8(5,3)  10(5,5)  6(4,2)  5(3,2)  9(3,6)  7(2,5)  3(1,2)  10(5,5)

5(3,2)  7(4,3)  10(6,4)  6(3,3)  8(6,2)  10(4,6)  10(5,5)  12(6,6)  4(2,2)  4(1,3)

8(5,3)  8(3,5)  7(3,4)  7(4,3)  12(6,6)  10(5,5)  5(1,4)  7(1,6)  6(2,4)  11(6,5)

8(2,6)  7(4,3)  10(4,6)  5(4,1)  3(2,1)  4(1,3)  7(6,1)  6(1,5)  3(2,1)  8(3,5)

9(6,3)  5(4,1)  9(6,3)  5(3,2)  10(4,6)  4(2,2)  6(2,4)  4(1,3)  8(6,2)  10(6,4)

7(6,1)  8(3,5)  6(1,5)  5(3,2)  8(5,3)  6(4,2)  9(6,3)  2(1,1)  4(1,3)  7(6,1)

7(5,2)  7(3,4)  7(1,6)  3(2,1)  6(4,2)  11(5,6)  10(4,6)  5(3,2)  8(2,6)  6(4,2)

8(5,3)  5(1,4)  3(2,1)  2(1,1)  11(6,5)  9(6,3)  11(6,5)  7(1,6)  10(4,6)  9(3,6)

10(4,6)  8(3,5)  9(6,3)  4(3,1)  8(3,5)  5(2,3)  2(1,1)  5(4,1)  7(4,3)  6(5,1)

8(4,4)  7(1,6)  10(5,5)  9(5,4)  3(1,2)  7(4,3)  5(1,4)  6(5,1)  8(2,6)  5(4,1)

Calculation:

1. The code is written in Java Language. This code gives the combinations for 100 dice rolls using random function. The code also calculates the frequency of each sum.

 

public class Dice {

 

public static void main(String[] args) {

int[][] a = new int[10][10];

int x,y;

int[] counter = new int[11];

 

for(int i = 0; i < 11; i++) {

counter[i]=0;

}

for(int i = 0; i < 10; i++) {

for(int j = 0; j < 10; j++) {

x=randInt();

y=randInt();

a[i][j]=x+y;

System.out.print(a[i][j]+”(“+x+”,”+y+”)  “);

counter[x+y-2]=counter[x+y-2]+1;

if(x>y) {

counterDiceOne++;

}

else if(x<y) {

counterDiceTwo++;

}

else draw++;

}

System.out.println(“”);

}

for(int i = 0; i < 11; i++) {

System.out.println(“Frequency of “+(i+2)+ “: “+ counter[i]);

}

 

}

private static int randInt() {

int rand= (int)(Math.random()*((6-1)+1)+1);

return rand;

}

}

2.rate of occurrence =