Jinjie Liao
Wednesday, October 23
Play With Die :
Probability of Dice
Abstract
In this experiment, we are going to find out what is the probability or the frequency that the combinations of 7 will occur with two dice. The result shows that the probability of getting the different combinations of seven is similar, but not the same due to outside forces. These results confirm my hypothesis because the difference between the expected result to the experimental result is approximately 0.06. This slight difference might be caused by errors during the experiment.
Introduction
The first dice were made by sheep knucklebones in Egypt around 5000 BC. We found dice similar to today’s dice, from Iraq, around 3000 BC (Carr). People usually made it for gaming. A dice with six sides has an equal probability of ⅙ of getting each number If we add one more dice and rolling both dice 100 times, then the probability to get the combinations of seven will add up to ⅙ in same frequency because there are 6 combinations of seven out of 36 total combination, such as (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). In this experiment, we are going to ignore the order of the numbers.
It is impossible to re-mark a pair of 6-sided dice so that the possibilities of sums 2, 3, … 12 are the same. But it is possible to mark 7…so that the sum of 7… appears equally (Bermudez, Medina, Rosin, Scott (2013)). In other words, the combinations of seven should have the same frequency outcome. To demonstrate that the frequency of different combinations of 7 is equally likely, the two dice will be tested.
Materials and Methods (Procedure)
One hundred trials of rolling dice are set up; require rolling dice with the same person at the same place; the dice will not be replaced. The specific step is:
Step 1: Get two dice with the same size and mass
Step 2: Find a good space to roll the dice
Step 3: Rolling two dice at the same time for 100 times
Step 4: Record the data
Step 5: Organize the data
Step 6: Find the frequency of the combinations that have occurred is equal to 7
Step 7: Compare the probability
Results
- The frequency that 7 will have occurred on 100 trials was 23/100, which is equal to 0.23.
- The frequency of 7 that will occur if the probability of each number is equally likely ⅙ and it is approximately about 0.1667.
- The frequency to get (1,6) was: 9/100, which is equal to 0.09.
- The frequency to get (2,5) was: 8/100, which is equal to 0.08.
- The frequency to get (3,4) was: 7/100, which is equal to 0.07.
The result shows that the frequency of getting the probability of (1,6), (2,5), and (3,4) is decreasing by 0.01. Table 1 shows the outcomes and the sum of the outcomes in the trials that get seven. We can see there are four trials in red was showing the same outcome four times. It is a very low chance to get the same outcomes on nearly 2 to 10 trials. So I think there might be some errors. Table 2 shows all the combinations of two dice, and we can see the probability to get 7 is ⅙.
In order of the frequency from greatest to lowest:
0.09 > 0.08 >0.07 (1,6) > (2,5) > (3,4)
0.23 > 0.1667 frequency in 100 trials to get 7 > probability to get 7
0.23 – 0.1667 = 0.0633 the difference between the test and expected variables
Table 1. The results of two dice in 100 times
| Trials | Dice 1 | Dice 2 | Result |
| Trials 6 | 1 | 6 | 7 |
| Trials 8 | 1 | 6 | 7 |
| Trials 16 | 2 | 5 | 7 |
| Trials 17 | 3 | 4 | 7 |
| Trials 24 | 3 | 4 | 7 |
| Trials 33 | 6 | 1 | 7 |
| Trials 36 | 3 | 4 | 7 |
| Trials 38 | 6 | 1 | 7 |
| Trials 44 | 6 | 1 | 7 |
| Trials 49 | 2 | 5 | 7 |
| Trials 53 | 6 | 1 | 7 |
| Trials 54 | 5 | 2 | 7 |
| Trials 55 | 3 | 4 | 7 |
| Trials 58 | 1 | 6 | 7 |
| Trials 63 | 3 | 4 | 7 |
| Trials 67 | 3 | 4 | 7 |
| Trials 68 | 1 | 6 | 7 |
| Trials 72 | 2 | 5 | 7 |
| Trials 73 | 2 | 5 | 7 |
| Trials 86 | 2 | 5 | 7 |
| Trials 88 | 2 | 5 | 7 |
| Trails 90 | 1 | 6 | 7 |
| Trials 94 | 2 | 5 | 7 |
| Total:23 |
Table 2. Different combination results
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
Analysis
In this experiment, the frequency of each combination of 7 is very similar. However, there are slight differences of 0.01 ~ 0.02 percentage caused by the errors and it might result in the outcome. First, the dice might have some problems that we didn’t know about. Second, the environment changing, such as the dice touching other materials and then stopping. The last is the human error of recording data and calculating. However, my hypothesis is believable because there are only about 0.06 differences between the test and the expected variable. According to Are Stupid Dice Necessary?, by Frank, B, “When p = 2 and q = 3, we have a standard 6-sided dice. If we want a fair dice, it makes sense to limit ourselves to common face entities…” (Bermudez, et al (2013)) This means the probability of the combination in dice will change if we add more dice or face of the dice. But from the previous evidence quoted in the introduction, the probability of seven is kept the same. Therefore, the probability of dice on the game table to win is not the same as the probability of dice on the game table to loss, and the frequency of an outcome of seven should be similar.
Conclusion
I am satisfied with the result of the frequency to get the combination of 7 with 0.09 for (1,6), 0.08 for (2,5) and 0.07 for (3,4). It is important to learn this because, from the probability of the dice, we can extend to many combinations with the face and number of the dice. During the experiment, I found many insufficient for my procedure. Next time, I would like to try multiple dice for 100 trials and compare their results. It can help provide better and stronger support for my hypothesis. I wonder if we add more faces to the die, does the probability of two dice getting 7 change or not?
Work Cited List
Frank, B., Anthony, M., Amber, R. and Eren, S. (2013, September) Are Stupid Dice Necessary?.
Mathematical Association of America, September, 2013, from
Carr, K.E. (2019, May 7) History of dice – When were dice invented?. Retrieved October
19, 2019, from https://quatr.us/west-asia/dice-invented-history-dice.htm.
Appendix
| Trial 2 | 4 | 5 | 9 |
| Trial 3 | 4 | 5 | 9 |
| Trial 4 | 6 | 2 | 8 |
| Trial 5 | 1 | 1 | 2 |
| Trial 6 | 1 | 6 | 7 |
| Trial 7 | 6 | 2 | 8 |
| Trial 8 | 1 | 6 | 7 |
| Trial 9 | 3 | 6 | 9 |
| Trial 10 | 4 | 6 | 10 |
| Trial 11 | 4 | 5 | 9 |
| Trial 12 | 3 | 5 | 8 |
| Trial 13 | 1 | 1 | 2 |
| Trial 14 | 1 | 2 | 3 |
| Trial 15 | 1 | 3 | 4 |
| Trial 16 | 2 | 5 | 7 |
| Trial 17 | 3 | 4 | 7 |
| Trial 18 | 4 | 4 | 8 |
| Trial 19 | 6 | 5 | 11 |
| Trial 20 | 6 | 2 | 8 |
| Trial 21 | 6 | 5 | 11 |
| Trial 22 | 3 | 6 | 9 |
| Trial 23 | 3 | 6 | 9 |
| Trial 24 | 3 | 4 | 7 |
| Trial 25 | 1 | 4 | 5 |
| Trial 26 | 6 | 5 | 11 |
| Trial 27 | 1 | 2 | 3 |
| Trial 28 | 3 | 1 | 4 |
| Trial 29 | 1 | 2 | 3 |
| Trial 30 | 6 | 2 | 8 |
| Trial 31 | 1 | 2 | 3 |
| Trial 32 | 6 | 6 | 12 |
| Trial 33 | 6 | 1 | 7 |
| Trial 34 | 6 | 5 | 11 |
| Trial 35 | 3 | 5 | 8 |
| Trial 36 | 3 | 4 | 7 |
| Trial 37 | 6 | 2 | 8 |
| Trial 38 | 6 | 1 | 7 |
| Trial 39 | 4 | 6 | 10 |
| Trial 40 | 2 | 4 | 6 |
| Trial 41 | 6 | 2 | 8 |
| Trial 42 | 4 | 2 | 6 |
| Trial 43 | 4 | 1 | 5 |
| Trial 44 | 6 | 1 | 7 |
| Trial 45 | 3 | 3 | 6 |
| Trial 46 | 3 | 2 | 5 |
| Trial 47 | 4 | 2 | 6 |
| Trial 48 | 1 | 1 | 2 |
| Trial 49 | 2 | 5 | 7 |
| Trial 50 | 4 | 5 | 9 |
| Trial 51 | 2 | 4 | 6 |
| Trial 52 | 2 | 2 | 4 |
| Trial 53 | 6 | 1 | 7 |
| Trial 54 | 5 | 2 | 7 |
| Trial 55 | 3 | 4 | 7 |
| Trial 56 | 3 | 6 | 9 |
| Trial 57 | 3 | 2 | 5 |
| Trial 58 | 1 | 6 | 7 |
| Trial 59 | 6 | 4 | 10 |
| Trial 60 | 1 | 4 | 5 |
| Trial 61 | 2 | 2 | 4 |
| Trial 62 | 3 | 3 | 6 |
| Trial 63 | 3 | 4 | 7 |
| Trial 64 | 3 | 6 | 9 |
| Trial 65 | 1 | 1 | 2 |
| Trial 66 | 1 | 4 | 5 |
| Trial 67 | 3 | 4 | 7 |
| Trial 68 | 1 | 6 | 7 |
| Trial 69 | 1 | 4 | 5 |
| Trial 70 | 1 | 5 | 6 |
| Trial 71 | 6 | 4 | 10 |
| Trial 72 | 2 | 5 | 7 |
| Trial 73 | 2 | 5 | 7 |
| Trial 74 | 3 | 6 | 9 |
| Trial 75 | 6 | 3 | 9 |
| Trial 76 | 2 | 2 | 4 |
| Trial 77 | 6 | 4 | 10 |
| Trial 78 | 6 | 3 | 9 |
| Trial 79 | 1 | 4 | 5 |
| Trial 80 | 1 | 2 | 3 |
| Trial 81 | 1 | 1 | 2 |
| Trial 82 | 6 | 4 | 10 |
| Trial 83 | 2 | 4 | 6 |
| Trial 84 | 5 | 5 | 10 |
| Trial 85 | 1 | 5 | 6 |
| Trial 86 | 2 | 5 | 7 |
| Trial 87 | 1 | 4 | 5 |
| Trial 88 | 2 | 5 | 7 |
| Trial 89 | 2 | 3 | 5 |
| Trial 90 | 1 | 6 | 7 |
| Trial 91 | 1 | 4 | 5 |
| Trial 92 | 6 | 2 | 8 |
| Trial 93 | 3 | 3 | 6 |
| Trial 94 | 2 | 5 | 7 |
| Trial 95 | 2 | 3 | 5 |
| Trial 96 | 5 | 5 | 10 |
| Trial 97 | 5 | 3 | 8 |
| Trial 98 | 5 | 5 | 10 |
| Trial 99 | 6 | 3 | 9 |
| Trial 100 | 6 | 2 | 8 |
