lab report

                     

 

 

 

                  Roll the Dice!

Probability of the Highest Sum of Two Dice 

           Kimberly Chicaiza

                   3/8/23

Abstract:

This lab report was created to experiment with the probability of the highest sum of two dice rolled. To make these statistics, two dice are rolled at the same time. The sum of the two dice is up to 2 through 12. It is recorded on a sheet the amount of times it rolls that same addition in two trials of 100 times rolled. What probability of sum do you think from 2 through 12 will be the most constant? In order to know the highest sum, I gathered the two numbers from each dice and added them together with its total. My hypothesis is that the greatest sum will be 7 and the lowest sum will be 2 and 12, due to its low probability of rolling that exact sum.

Introduction:

Has probability ever been hard for you to understand? Well, rolling dice is the best opportunity to learn about statistics. Similarly, like the casinos, how do these lucky winners know when to bet on choosing the right number? Well, probability plays a big role when emphasizing what will most likely happen by choosing the exact numbers in many ways, since there are double 6 faces and chances. The formula of probability is probability of both = Probability of outcome one × Probability of outcome two. There are 36 possible outcomes. It determines that the sum of 7 has the highest chance of appearing because it has a 16.67% probability due to the number of lists. 

Materials:

  1. Two dice
  2. Pen and paper/ computer skills to record data
  3. Calculator
  4. Hard Work

Methods:

  1. In order to elaborate this experiment, there are many steps to conclude the understanding of probability by rolling two pairs of dice. 
  2. First, a hypothesis is created from your own perspective and intuition or experience. There are two different ways to do this experiment, such as, actually rolling dice or using a simulation to make it more easy. Choosing a dice rolling simulation, codes are used to roll each dice 100 times.
  3.  Next, record your database on Google Sheets or Excel the amount of times it rolled that same sum constantly. Through the help of this table you create a chart for an abstract observance to visualize which product is repetitive. 
  4. Then, research based on Dice Probability to inform basic information of probability and guide your hypothesis. 
  5. Lastly, compare your database to your hypothesis.   

Results:

After the simulation rolled 100 times on each dice, the table as shown in Figure 1 explicitly demonstrates which sum was constantly produced. As well,  Figure 2 shows an abstract visualization where 8 and 7 were the greatest outcome.

 

Outcome Amount of Sum of Two dice
2 2
3 4
4 10
5 7
6 10
7 16
8 20
9 9
10 14
11 4
12 3

Figure 1- Table of the Amount of Outcome Repetition

 

Figure 2- Abstract Chart, an explicit view to know which sum was constantly rolled.

 

Analysis:

To summarize, my hypothesis is that I predicted the probability that the sum of 7 would be the highest sum because it has a variety of combinations. Also, the probability of the lowest sum is 2 and 12 because it has a very low variation from all the 36 outcomes of two dice rolled. According to the Probability Formula, the sum of seven had the highest percentage out of all the other sums. Figures 1 and 2 show that the sum of seven appears constantly. Also, the sum of 2 and 12 as shown in Figure 2 explicitly demonstrates that it is the lowest sum rolled out of 100 times. Figure 3 reveals that 7 has the most list combination rather than 2 and 12.

 

Outcome List of Combinations Total
2 1+1 1
3 1+2, 2+1 2
4 1+3, 2+2, 3+1 3
5 1+4, 2+3, 3+2, 4+1 4
6 1+5, 2+4, 3+3, 4+2, 5+1 5
7 1+6, 2+5, 3+4, 4+3, 5+2, 6+1 6
8 2+6, 3+5, 4+4, 5+3, 6+2 5
9 3+6, 4+5, 5+4, 6+3 4
10 4+6, 5+5, 6+4,  3
11 5+6, 6+5 2
12 6+6 1

Figure 3- List of Combination for outcomes 1-12

Through much research into visual experiments on probability, a similar study appeared that did the same method of rolling a dice simulation. On the other hand, it used two different outcomes, such as 9 and 10. The experiment was to find out which sum is highly produced. According to their hypothesis, the sum of 10 has a higher probability than the sum of 9. The statement introduced the hypothesis that “There are more possibilities for the sum10 than for the sum 9 and this is the reason why the sum 10 occurs more frequently than the sum 9 in a large number of repeated random experiments.” (LUKAC 38). The table shown in Figure 3 demonstrates that the sum of 10 had a higher frequency than the sum of 9.

 

3 COUNT 11
4 COUNT 67
5 COUNT 145
6 COUNT 227
7 COUNT 345
8 COUNT 510
9 COUNT 561
10 COUNT 692
11 COUNT 601
12 COUNT 556
13 COUNT 501
14 COUNT 354
15 COUNT 214
16 COUNT 133
17 COUNT 60
18 COUNT 23

Figure 4-The frequency of individual sums.

Conclusion:

To finalize the experiment, I can’t say that it is easy but not impossible to learn about probability, as it takes one day at least to comprehend. But, it was a fun experience because you’re able to play with dice and compare your thoughts with actual evidence. As shown throughout the experiment, there are many possibilities to sum a specific number by using probability; it will give a greater chance to know which number is constantly summed by rolling two dice. I was able to conclude my hypothesis was correct because, as shown in figure 1 and 2, the highest sum was seven and the least sum was 2 and 12. Therefore, probability plays a big role in our daily life as we evolve and become more independent, because it helps us choose better options to thrive in life and be a successful person. That is how the lucky winner in casinos knows the knowledge of probability. 

Citation:

Dice Simulation: https://replit.com/@KimberlyChicaiz/LongtermWeepyConditions 

 

(“How to Calculate Dice Probabilities?”) https://www.toppr.com/guides/maths/probability/how-to-calculate-dice-probabilities/#Solved_Question_For_You.

 

LUKAC, S.; ENGEL, R. Investigation of probability distributions using dice rolling simulation. Australian Mathematics Teacher, [s. l.], v. 66, n. 2, p. 30–35, 2010. Disponível em: https://search-ebscohost-com.ccny-proxy1.libr.ccny.cuny.edu/login.aspx?direct=true&db=a9h&AN=51533174&site=ehost-live . Acesso em: 12 mar. 2023. 

 

Appendix:

Code used for Dice Rolling Simulation:

print( “Dice 1: “)

import random

i=0

while i !=100:

    for i in range(1,101):

      y=random.randint(1,6)

      print (i,”:”, y)

print()

print( “Dice 2: “)

i=0

while i !=100:

  for i in range(1,101):

      x=random.randint(1,6)

      print (i, “:”, x)

 

Table of 100 Number of Rolls with Number of Outcome:

 

# of times rolled Dice 1: Dice 2: Sum
1 3 1 4
2 6 3 9
3 1 1 2
4 2 2 4
5 5 4 9
6 1 4 5
7 6 2 8
8 2 6 8
9 4 3 7
10 2 1 3
11 4 3 3
12 2 5 7
13 4 4 8
14 4 4 8
15 5 3 8
16 2 4 6
17 3 6 9
18 6 6 12
19 4 4 8
20 6 4 10
21 6 3 9
22 4 6 10
23 3 1 4
24 1 3 4
25 3 4 7
26 2 3 5
27 3 1 4
28 5 3 8
29 5 3 8
30 6 1 7
31 3 1 4
32 4 3 7
33 1 6 7
34 2 3 4
35 2 6 8
36 6 4 10
37 6 4 10
38 3 5 8
39 2 2 4
40 6 6 12
41 3 3 6
42 2 6 8
43 4 2 6
44 5 2 7
45 3 1 4
46 1 1 2
47 5 5 10
48 4 2 6
49 3 1 4
50 2 6 8
51 5 3 8
52 6 4 10
53 5 5 10
54 4 2 6
55 4 6 10
56 6 2 8
57 1 6 7
58 6 1 7
59 4 5 9
60 4 5 9
61 6 5 11
62 5 6 11
63 3 6 9
64 5 3 8
65 6 4 10
66 2 5 7
67 2 3 5
68 4 5 9
69 5 1 6
70 4 1 5
71 3 3 6
72 5 1 6
73 2 5 7
74 2 4 6
75 3 2 5
76 2 4 6
77 1 2 3
78 4 6 10
79 6 6 12
80 6 2 8
81 4 6 10
82 5 6 11
83 5 5 10
84 3 5 8
85 6 4 10
86 5 3 8
87 6 2 8
88 4 4 8
89 4 3 7
90 6 4 10
91 2 1 3
92 6 5 11
93 2 5 7
94 2 3 5
95 1 4 5
96 6 6 12
97 5 2 7
98 2 5 7
99 1 6 7
100 3 6 9