Just How Do the Dice Really Fall?

I teach math. One of the topics I cover in a math class that is geared for liberal arts is basic probability. I like to show students how games of chance are not "lucky" but are actually somewhat predictable. Here is how I approach the fact that "lucky sevens" aren't really derived from luck.

I start with a chart showing all the different pairings we can get when we roll a pair of dice. The pairs start with rolling a two and end with rolling a twelve. We look at die 1 and die 2. If the same number is rolled on each die, we only consider it that one time, but if I roll a three on die 1 and a four on die 2, it is considered different from a four on die 1 and a three on die 2.

Here is a rough view of what I give them.

There is only one way to roll a 2: (1, 1)
There are two ways to roll a 3: (1, 2) and (2, 1)
There are three ways to roll a 4: (1, 3), (2, 2), and (3, 1)
There are four ways to roll a 5: (1, 4), (2, 3), (3, 2), and (4, 1)
There are five ways to roll a 6: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1)
There are six ways to roll a 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1)
There are five ways to roll an 8: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2)
There are four ways to roll a 9: (3, 6), (4, 5), (5, 4), and (6, 3)
There are three ways to roll a 10: (4, 6), (5, 5), and (6, 4)
There are two ways ro roll an 11: (5, 6) and (6, 5)
There is only one way to roll a 12: (6, 6)

This gives us a chart showing the way the dice can fall. If you notice, there are more combinations that will give us a SEVEN than any other number. In this way, we can see that our probability of rolling a seven is higher than rolling any other number -thus sevens are not lucky, they just have a higher probability of being rolled.

I then have students to take a pair of dice and roll them 25 to 50 times. They work in pairs; one person rolls the dice and the other person records the data. Then they switch roles. Once both partners have had a chance to complete 25 to 50 dice rolls, they complete a frequency distribution for their data and compare it to the chart I have presented at the beginning of the lesson.

Once we look at the data of each pair of students, we compile the data for the entire class. We then make a comparison of this data with our chart from the beginning of class. We usually find that the data from individual students will vary a good bit from the chart that I present at the beginning of class, but as we add data to it for the entire class, the data begins to look more and more like the chart of possibilities for rolling the dice.

At the end of class, I show the class that if we look at my original chart (from the beginning of class), we will find that there are 36 ways our pair of dice can land. Thus, 36 is the bottom of the fraction that we will use to figure the theoretical probability of rolling each individual number. Since there are 6 ways to roll a 7, the probability of rolling a 7, noted as P(7) is equal to 6 divided by 36 or 6/36. We can convert this to a decimal. We do the same for each of the other numbers and get our theoretical probability.

I then make the assignment for each student to find probabilities of rolling each number based on their data alone, their data combined with their partner's data, and the class data as a whole. During the next class, we discuss the differences between each probability that we get.

I hope your students enjoy the hands-on lesson on probability as much as mine always do. I think it is always good to try and incorporate lessons into the curriculum that will help students really understand why it is important that they learn a certain topic.