Finding the Shape of Chaos:
Studying Outcome Probability Using a Six-Sided Pair of Dice.
Rene Diaz-Rocha
English 20107 Section L
10/23/18
Abstract:
We can’t know the precise outcome of a random event with certainty. However, the outcome of a random event has a precise likelihood of occurring. In this study I analyzed the probability of numerical outcomes when rolling a pair of six-sided dice. I explored the relationship between the quantity of trials and the probability of specific numbers to occur during a roll. I concluded that the data obtained resembles the calculated probability of each outcome more closely with increasing number of trials. As expected, seven was the most frequent outcome when rolling two dice. Moreover, a tally of individual dice outcomes indicates that each value from one to six is just as likely to occur as any other value. This experiment is an effective way to introduce principles of probability to students.
Introduction:
Have you ever wondered why it is so hard to win the lottery? The lottery numbers are chosen at random and the odds of guessing those numbers are extreme small. The probability of winning the lottery is the total number of winning lottery numbers divided by the t
otal number of possible lottery numbers. In other words, the number of ways of picking items for a very large set of items. As Ron
Lancaster, Professor at the Ontario Institute for Studies in Education of the University of Toronto, explains: “Suppose there are 1 to 49 different numbers to choose from in 7 possible configurations: Your chances of winning are 0.00000116%.” (page 572). Mr. Lancaster shows that the odds of predicting an outcome are determined by the possible outcomes of an event. Rolling a pair of dice is the simpler
way to illustrate this concept. In a six-sided dice there are six possible numerical outcomes: (1), (2), (3), (4), (5), (6). Because rolling a single dice produces only one outcome, each number has one out of six chances of occurring. When we consider the sum of dice A and dice B, we have 11 possible aggregate outcomes: (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12). There is only one way of rolling a (2): (A1+B1).
There’s two ways of rolling a (3): (A2+B1) and (A1+B2). There are three ways of rolling a (4): (A2+B2), (A1+B3), and (A3+B1). There are four ways of rolling a (5): (A2+B3), (A3+B2), (A1+B4), and (A4+B1). There are five ways of rolling a (6): (A1+B5), (A5+B1), (A4+B2), (A2+B4), and (A3+B3). There are six ways of rolling a (7): (A1+B6), (A6+B1), (A5+B2), (A2+B5), (A3+B4), and (A4+B3). There
are five ways of rolling an (8): (A2+B6), (A6+B2), (A3+B5), (A5+B3), and (A4+B4). There are four ways of rolling a (9): (A3+B6), (A6+B3), (A4+B5), and (A5+B4). There are three ways of rolling a (10): (A4+B6), (A6+B4), and (A5+B5). There are two ways of rolling (11): (A6+B5) and (A5+B6). Finally, there is only one way of rolling (12): (A6+B6). We can observe that there are more configurations, out of the 36 possible configurations, that add up to (7). I therefore hypothesize, when rolling a pair of dice 100 times, (7) will be the most frequent outcome.
Materials:
1. Two green six-sided dice. The entire surface area of each dice is 10.14 squared centimeters. The area of each side of each dice is 1.64 squared centimeters.
2. A light brown rectangular wooden table with dimension 1.2 X .9 squared meters.
3. A red dry erase marker used to draw a line on the wooden table.
4. A sheet of paper with dimensions 8.5 X 11 inches that I used to tally the results of each dice roll
5. A #2 mechanical pencil used to write down the results of each dice roll.
Procedure:
1. Draw a line in the middle of the table: one side of the table is labeled A and the other side B.
2. I roll the pair of dice onto the wooden table. I try to do it so that each die ends up in a separate side of the table. I repeat this process 100 times.
3. I make one tally with numbers 1 through 6 to keep track of the frequency of outcomes for each die.
4. I make one tally with numbers 2 to 12 to keep track of the frequency of outcomes for the sum of two dice (I’ve chosen to use a hand-written tally because I can see a gradual change in the shape of the tally as I continued to roll the pair of dice trial after trial).
5. I transfer the information of the tallies for 100 trials into Table 1 (see Appendix I).
6. I used the information from my data table to generate a bar graph that illustrates the relationship of each outcome to its frequency for analysis.
Results:
Figure 1 shows the frequency of outcomes when rolling two dice 100 times. The order of outcomes from the most to the least frequent outcomes in our experimental data are as follows: (7)>(8)>(6)>(9)>(5)>(4)>(10)>(11)>(3)>(12)=(2). This can be interpreted as the probability distribution of the event for 100 trials, because the frequency of the event determines the chances of that event occurring.
Figure 2
shows the frequency of individual dice outcomes. This can be interpreted as the probability distribution of each individual dice outcome for 200 trials. The order of outcomes from the most to the least frequent outcomes in our experimental data are as follows: (2)>(6)>(5)>(3)>(1)>(4).
Analysis
My results indicate that (7) is the most frequent outcome when rolling a pair of dice 100 times. A pattern is clear in figure 1 (Rolling 2 Dice: Trial 1-100): as we move farther to the left or right of (7) the frequency of each outcome decreases. We can conclude from our results that the outcome depends on the sum of two separate outcomes of each dice. Therefore, the probability of an outcome of two dice is directly proportional to the amount of combinations of individual dice outcomes that would add up to that number. For example, the outcome (7) has six possible combinations: (A1+B6), (A6+B1), (A5+B2),(A2+B5), (A3+B4), (A4+B3). This is the highest amount of combinations out of all possible outcomes. Therefore, the experimental results are consistent with our predictions. The outcomes (6) and (8) each have five possible combinations each: (A1+B5), (A5+B1), (A4+B2), (A2+B4), (A3+B3) for the outcome (6), and (A2+B6), (A6+B2), (A3+B5), (A5+B3), (A4+B4) for (8). However, our experimental data shows the outcome (8) to be more frequent than (6). This shows the non-deterministic nature of probability. The fact that both numbers have the same number of combinations that add up to them doesn’t imply that they will occur at the same frequency, rather that they are likely to appear in the same frequency.
Another implication, from the collected data, is that if we pick a small number of trials from our total number of trials the data would not resemble our predicted probability distribution. From trial 11 to trial 19 (see Appendix I Table 1, page 9) the outcomes are as follows: (8), (9), (6), (4), (10), (6), (7), (8), (5). In a small sample like this one, (6) and (8) are the most frequent outcomes. This exemplifies another
characteristic of probability; experimental data is more ideal with in a larger sample. A study by Colin Foster and David Martin, both Professors at The School of Education at The University of Nottingham, reinforces my findings. In their article “Two-Dice Horse Race” they examine the results of a game involving the outcome of the sum of two dice. The frequency of each outcome represents a distance a
horse can advance. Thus, the most frequently occurring number will win the race. They present following conclusion: “The explanation for horse 7’s winning performance in terms of combinations of numbers that sum to 7 is only a small part of the story; the length of the track is also important. From a pedagogical point of view, although using this task to introduce sample space diagrams might be helpful, great care in follow-up questions is needed. To use the two-dice horse race to 10 spaces merely to illustrate the results of a single throw of two dice may be unhelpful. Simple-sounding dice scenarios can be much more complicated than they might at first sight appear”. (page 100)
When taken separately,the component rolls of dice A and dice B amount to 200 outcomes. Analysis of figure 2 (Individual Dice
Outcomes: Trial 1-100) indicates the lack of a pattern like the one in figure 1. The reason is each side of the dice has the same probability of appearing. My experiment has important educational applications. Lionel Pereira-Mendoza is Professor at the National Institute of Education, Nanyang Technological University, Singapore. In his article, “Using Dice: From Place Value to Probability”, he illustrates important applications of a similar experiment as follows: “A die is selected and is tossed sixty times. The player keeps a record of the faces and this is used as a basis for discussing the probability of a particular number or shape appearing. By using two whole number dice and keeping a record of the sum, this activity can be used as an introduction to the idea of normal distribution.” (page 12) My experiment closely resembles the methodology set forth by professor Pereira-Mendoza. In this manner, my experiment highlights probability distribution concepts
in a simple accessible way; it’s an efficient educational tool.
Conclusions
This experiment illustrates the importance of collecting data from large set of events to stablish the likelihood for those events to occur. This is a fundamental principle of probability, which can be applied to a vast number of events, from winning the lottery to avoiding car accident. Probability helps us grasp the chaotic and random nature of everyday life. Understanding, why some events are more probable than others is an important analytical skill. In this experiment we saw why the outcome (7) is more likely to occur by finding a relationship of the numerical combinations in two dice that add to an outcome and that outcome. Like the rolling of a pair of dice, people all over New York take chances, whether you are applying for a job or finding a good restaurant to eat, collecting information about these events can help
you find success.
Works cited
1.Foster, C, & Martin, D. ( September 2016). Two Dice Horse Race. Teaching Statistics, Vol. 38, No3, pp. 98-101. Published by: Teaching Statistics Trust. Stable URL:https://web-a-ebscohost-com.ccny-proxy1.libr.ccny.cuny.edu/ehost/pdfviewer/pdfviewer?vid=3&sid=c2700bcc-d573-4617-a4f9-514152f3cfc8%40sessionmgr4007
2. Lancaster, R. (April 2013). Lotto Prize.1//Lotto Prize.2. The Mathematics Teacher, Vol. 106, No. 8(April 2013), pp. 570-572. Published by: National Council of Teachers of Mathematics. StableURL: https://www.jstor.org/stable/10.5951/mathteacher.106.8.057
3. Pereira-Mendoza, L. (April 1981). Using Dice: From Place Value to Probability. The Arithmetic Teacher, Vol. 28, No. 8 , pp. 10-12 Published by: National Council of Teachers of Mathematics. Stable URL:https://www.jstor.org/stable/41191861
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