May the Odds Be in Your Favor:
The Reality of Probability
Abstract
Probability is defined to be the percentage for a chance that something might happen. For example, a 100% probability that Tajinder Singh will be getting an A in his writing for engineer’s class means that this result will be certain, and no other outcomes exist. Although we might think of probability to be another topic in our Statistics class that we took in High School or College, it is a very useful concept that has consumed many fields and is widely used. We begin to learn this concept through an experiment using pair of die. After rolling the two die, we add the numbers. We repeat the process multiple times and analyze the results. After analysis, the results show that there is greatest probability for the summation of two numbers from the two dice to equal to seven, eight, and nine. Amongst these numbers, eight has the greatest probability if occurrence.
Introduction
If you have ever played cards against a really good player, you might be scratching your head throughout the whole game trying to understand how your opponent is winning. A simple game that relies heavily on luck and still you are unable to defeat this one person. However, this simple game of cards is not as simply as one might think of it to be. It requires some luck, but a lot of knowledge about probability. One of the most fundamental aspects that a player must learn in order to play cards and win is understanding the concept of probability. And not only will understanding probability help you climb the ranks and become a professional cards player, but you must also be able compute probabilities quickly and constantly as the game goes on. Professional card players are so trained in probability that they are doing the calculations in their heads within seconds along with planning their next move. Therefore, playing cards is not simply luck, but mathematics. For those who are not aware of probability, “it is the proper existing tool for managing uncertainty.”(Castagnoli5) For example, if you have ever looked at the weather before heading out to work, school or anywhere else, you might have seen that rain is often represented with a percentage. That percentage is showing the probability for rain to occur during a specific time. For example, there is 30% probability for New York to experience rain on Monday at 3:00 pm. We are surrounded by probabilities. Even for people who are investors and interested in stocks, you might want to understand probability to get the most out of your investments. However, we will only discuss the concept of probability through a demonstration of rolling of a pair of die and adding the two numbers produced. I believe that the results of this experiment will not be uniform throughout. I will not be getting an equal quantity of results from 2 to 12. Instead, I will have a greater quantity for the number seven. Through this lab, we will understand the fundamentals of probability and how it is used through experimentation.
Materials and Methods
Let’s start out journey into statistics and probability by creating an experiment which will question the concept of probability and help to understand why it exists. One of the most common methods of analyzing probability is through flipping a coin. However, that is too simple so we will experiment probability through rolling a pair of dice and recording the summation of the numbers that they both land on.
Materials and Methods
Method/steps:
Results
Having completed the experiment by following the procedures above, we now begin to organize the data to better interpret the information that we have recorded. The data is placed in numerical order and each occurrence of number from 2 to 11 is counted. For better portrayal of the data found, the pie chart in figure 1 was created.
Figure 1: Pie Chart representing the occurrence of summation of two random numbers generated by the rolling of a dice.
Examining the chart above, we see that the number eight has the greatest probability of occurrence within our experiment with the percentage of 22%. This indicates that within a hundred rolls of a pair of dice, the summation of two dies will produce a number of eight about 22 times. Following the number eight, the number seven has the second highest probability of occurrence with 17% and following seven is the number nine with 16% probability. Summations lower than seven and higher than nine had lower probabilities.
Figure 2: shows the data recorded and portrays the bell curve that is very useful in statistics.
Analysis
As discussed above, we notice that there are greater numbers occurrences for the summation of seven, eight, and nine. And among these three, eight has the greatest probability of 22%. We have greater probabilities for seven, eight, and nine because there exist greater numbers on the number cube that can add to these three number than other numbers that are produced from summation. For example, the number seven can be produced by the summation of: three and four: five and two: and six and one. The number eight can be produced through the summation of: four and four: six and two: and five and three. And lastly, the number nine can be produced through: three and six: five and four: six and three. When these numbers which each have three options are compared to a lower probability number such as two, we notice that two has only one option for summation. That is the addition of one and one on the dice. Similarly, the number ten has only two orders of summation which are five with five, and six with four. The number 12 had not occurred once within our 100 trials. The probability of obtaining the number 12 is 1/36 which is equal to the probability of obtaining the number 2. Another analysis to be made is the shape of the bar graph in figure 2. Since these summations were randomly generated and certain numbers had greater probability of occurrence than other numbers, we should have achieved a bell curve which is very important in statistics and is widely used for portraying information. The bar chart in figure 2 shows the data that was gathered and also has a layout for the bell curve where probability of the numbers seven, eight, and nine were the greatest and the probability of the numbers decreased as you moved further away from these numbers.
My hypothesis had stated that the probability of the summation to equal to seven would be the greatest. However, after performing the experiment and graphing the results for visual analysis, my hypothesis was not completely correct. My results show that the summation of the number eight had the greatest probability followed by the number seven and then nine. These three numbers are relatively close for the probability of occurrence; eight have a probability of 22%; seven with a probability of 17%; nine with the probability of 16%. The probability for the occurrence for other numbers below seven and above 8 are lower. Similarly, Roger W. Johnson, author or Illustrating basic probability calculations using ‘craps’, states calculations for the probability of occurrence for certain numbers through the analysis of the game craps. He states, “As there are 36 equally likely outcomes when rolling a pair of dice, and eight of them correspond to getting a 7 or 11, the chance the shooter wins on his first throw is” 8/36, which can be simplified to 2/9. (Johnson98) However, if we only look at the probability for only seven, we would have the result 1/6 or about 17%. And we compare that to the probability of 11, we have 1/18 or about 6%. Using Johnsons example of the craps game, we can compare it to out experiment and realize that it is not too far off from the given probabilities. Since the rolling of dice and outcome of the numbers were random, we would not obtain exactly the same results, but a bell curve is formed for our bar graph which is crucial for this experiment.
Conclusion
From games such as cards and crap to looking at whether or finances, we are enveloped by probability and its applications. Thus, it is very important that we understand what these numbers represent and how to best interpret them for whatever purpose were need. This report looks at the fundamentals of probability through the application of an experiment. We roll a pair of dice and add up the outcomes to determine which summation from two to twelve has the greatest probability of occurrence. Using the information we had obtained and analyzing it, we see that the number seven, eight, and nine have the greatest probability. We can use these key concepts of probability to begin analyzing information that we are given on a day-to-day basis. Using this knowledge, we can begin to explain how probability plays a key role in our life.
Work Cited List:
Appendix:
| Summations: | Occurrences |
| 2 | 1 |
| 3 | 3 |
| 4 | 5 |
| 5 | 12 |
| 6 | 12 |
| 7 | 16 |
| 8 | 21 |
| 9 | 15 |
| 10 | 2 |
| 11 | 9 |
Organization of the data obtained from experiment
Graph from Roger W. Johnson’s illustration of probability through the game of craps. Shows the occurrence of each number (from 2-12) which can help to calculate the probability of each. There are 36 total outcomes.
