A Game of Die:

Probability and the Most Likely Outcome of Two Dice

Mukhlisakhon Fazliddin

October 30, 2018

 

Abstract:

In this experiment, we learn about probability by rolling dice and finding the most likely sum of the two dice. For the control group, we calculated the probability for each outcome by using a tree diagram. For the experimental group, we wrote a computer program that produces 100 sets of two random numbers and the sum of those two numbers. The most likely outcome was seven for both the calculated portion and the experimental portion, but in the experimental part, seven was more likely than in the calculated.

Introduction

Odds of being injured by a toilet are 1 in 10,000 (Center for Disease Control and Prevention, 2011). The purpose of this experiment is to understand basic probability by determining the most likely outcome of the sum of rolling two dice. It is interesting because while the probability of each outcome for rolling a die is the same, the probability for each outcome of rolling two dice varies. The outcomes of rolling two dice is most relevant in casino games but can be applied to other events, such a two dice horse race. We will look at the theoretical or mathematically calculated outcome and the experimental outcome. For my hypothesis, the most likely outcome will probably be seven because it is the middle number.

 

Materials

• pen/pencil
• paper
• computer/laptop, with an internet connection

 

Method

Part 1: Theoretical Outcome

1. Write out all the different possible outcomes, using a probability tree diagram.
2. Determine the probability of each possible outcome by counting how many times the number appears in the most outer branches of the tree diagram.
3. Record which had the highest probability of being the sum most rolled.

Part 2: Experimental Outcome

1.  Open Chrome, right click anywhere and click “Inspect.” At the top click on “Console.”
2. Write a function that calls for 100 sets of two random numbers from one to six and then sums up those numbers.

function roll(max) {

return Math.floor(Math.random() * Math.floor(max))+1;

}

for(var i = 0; i < 100; i++) {

dice1=roll(6), dice2=roll(6);

console.log(“[ ” + dice1 + ” ] [ ” + dice2 + ” ] = ” + (dice1+dice2) )

}

3. Tabulate the results by making one column all the possible sums and the second column the frequency of those sums. Make a bar graph.

 

Results

Part 1:

Figure 1 – table for the calculated  probability

for all possible sums

 

Figure 2 – pie chart for the percentages of the frequency of all the possible sums              

The pie chart displays the percentages of all possible outcomes of rolling two dice and shows that seven is the most likely outcome.

 

 

Part 2:

Figure 3 – table for the frequency of all the sums

Figure 4 – bar graph for the frequency of all the sums

Just as in part 1, part 2 shows that seven is the most likely outcome.

 

Analysis

The results for both the calculated version and the experimental version show that seven is the most likely outcome of the sum of rolling two dice. Seven is the most likely sum because most of the numbers on a die can add up to seven. However, in figure 4, one can observe that seven was rolled more in the experimental group than in the control group (figure 2). This shows that when it comes to probability, reality will not always be the same as the calculated probability. Colin Foster and David Martin (2016), in their work, “Two-dice horse race,” explain a game where two dice are rolled and horses 1-12 are allowed to move one step when its number is the sum of the two dice. Furthermore, they describe how increasing track length increases the chances of horse 7 winning the race (98-100). Similarly, in my experiment seven was rolled 22 times out of 100 whereas the theoretical only showed a 17 out of 100 of seven being rolled. Both experiments show that in real life seven will have a greater chance of being rolled if the number of total rolls increase.

Conclusion

Merriam-Webster dictionary defines probability as “the ratio of the number of outcomes in an exhaustive set of equally likely outcomes that produce a given event to the total number of possible outcomes.” We have found the most likely outcome of rolling two dice and my hypothesis is correct. Our winner is seven.

References

Center for Disease Control and Prevention (2011, June 10). Nonfatal Bathroom Injuries Among Persons Aged ≥15 Years — United States, 2008. Retrieved October 24, 2018, from https://www.cdc.gov/mmwr/preview/mmwrhtml/mm6022a1.htm?s_cid=mm6022a1_w

Foster, C., & Martin, D. (2016). Two-dice horse race. Teaching Statistics, 38(3), 98–101. https://doi.org/10.1111/test.12108

Merriam-Webster. (n.d.). Probability. Retrieved October 24, 2018, from https://www.merriam-webster.com/dictionary/probability