7.6.5: Ovid and Gambling

In this section we will consider a very old application of the idea of probability to the problem of rolling three fair, six-sided dice. Publius Ovidius Naso (43 BCE–18 AD), known in English as Ovid, was a Roman poet who lived during the reign of Augustus. Ovid was popular in Europe in the 12th and 13th centuries to the point that people composed works during that time and attributed them to Ovid (called pseudo-Ovidian works). De Vetula is a pseudo-Ovidian poem consisting of three books written in France in the mid-thirteenth century. The real author of the poem is unknown. It is written in the form of an autobiography of Ovid. The poem is a morality story in which the reader is led to transform his life from one centered on pleasure at the beginning to the one that is accepting the Christian faith by the end.

The first book describes Ovid’s youth, including pursuits such as hunting and fishing. The next section deals with reasons to avoid dice playing, and this is where probability and chance are addressed. A section of the poem deals with the calculation of the chances of various throws of dice. The section of the poem begins by describing the outcomes of one die and three dice. Of particular interest is the case when three dice are thrown and the numbers represented on the upper faces of the dice are added together. The author notes that certain sums occur more often than others. Specifically, with three dice, there are sixteen possible sums that can be observed. For example, while the outcomes 3, 4, 17, and 18 occur less frequently as there is only one configuration that will yield these sums, the outcomes 9, 10, 11, and 12 occur more frequently as there are six configurations that yield these sums.

Figure \(\PageIndex{1}\) shows a page from an early printing of the book. The top row of the table shown in the photograph shows (6,6,6), that is, each die rolled is a 6, which is the only possible way of observing a sum of 18. The next row shows (6,6,5), that is, one die is a 5, and the other two dice are each 6, is the only possible way of observing a sum of 17. When we look at the third row of the table, we get our first clue as to why some sums occur more often than others. In the case of observing a sum of 16 there are two possible configurations of the dice that give this sum: (6,6,4) and (6,5,5). The reasoning is now that because a sum of 16 can occur with two possible configurations of the dice, and there is only one possible configuration that will produce each of the sums 17 and 18, the sum 16 will be observed more often than either of the sums 17 or 18. Working down the table we can observe that the sums 9, 10, 11, and 12 should be observed the most often in that there are six configurations that will produce each of these sums. For example, the six configurations that will produce a sum of 10 are (6,3,1), (6,2,2), (5,4,1), (5,3,2), (4,4,2), and (4,3,3). However, as we shall see, this does not tell us the complete story.

Figure \(\PageIndex{1}\): A page from an early printing of De Vetula that shows why some sums of three dice occur more often than other (public domain image).

If you roll three dice many times and keep track of the sums, you will observe after a while that rolling a sum of 17 occurs more often than rolling a sum of 18, even though both sums only have one configuration that will produce the sum. The reason for this, as the author of De Vetula shows, is that for the sum of 18 all three dice must be equal to 6, while for the sum of 17 one of the dice is different than the other two. This makes a difference because we also need to consider which of the three dice is equal to 5. To see why this is true, suppose one of the dice is red, another is blue, and the third is green. Then there are three choices that will produce this configuration: the red die is 5 while the blue and green dice are 6, the blue die is 5 while the red and green dice are 6, and the green die is 5 while the blue and red dice are 6. In fact, as this argument shows, we should observe a sum of 17 about three times as often as we observe a sum of 18.

Figure \(\PageIndex{2}\) presents a photograph of a table from an early printing of De Vetula showing these differences. The first group of configurations will only occur once because all the numbers are the same. The second group of numbers can each occur in three different ways because two numbers are the same and one is different. The third, fourth, and fifth group of numbers have each dice equal to a different number, and the numbers are in order. Each of these configurations can occur in 6 different ways. For example, the configuration (1,2,3) can occur as (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).

Figure \(\PageIndex{2}\): A page from an early printing of De Vetula that shows the number of ways each configuration of three dice rolls can occur (public domain image).

In all there are \(6\times 6\times 6=216\) possible ways that three dice can be rolled that accounts for all the configurations and orderings. To compute the probability of a particular sum we need to count how many configurations and orderings correspond to that sum and divide by 26. Starting with the sum of 18, we find an easy case because there is only one configuration and one ordering, which is (6,6,6). Therefore, it follows that the probability of observing a sum of 18 is \(1/216\). The next largest possible sum is 17, which has one configuration but three possible orderings (6,6,5), (6,5,6), and (5,6,6). Therefore, it follows that the probability of observing a sum of 17 is \(3/216\). This process of counting continues for the remainder of the possible sums from 16 down to 3. Near the middle of the range, the counting procedure can become rather complicated. When all the counting is done, we end up with a table such as the one shown in the photograph in Figure \(\PageIndex{3}\), which is also from an early printing of De Vetula.

Figure \(\PageIndex{3}\): The table of probabilities for the sums of three dice from an early printing of De Vetula (public domain image).

In Figure \(\PageIndex{3}\) we can focus on the last column of the pictured table where the number of configurations and orderings, taken together, is reported. From this table we can observe that the sums of 18 and 3 will both occur with a probability of \(1/216\). Similarly, the sums of 17 and 4 will both occur with a probability of \(3/216\) and the sums of 16 and 5 will both occur with a probability of \(6/216\). What one quickly observes from the table is that the probabilities of observing a sum near the middle of the range are much higher, for example a probability of \(27/216\) for getting a sum of 10 or 11, when compared to the sums near the end of the range. The table is quite helpful in that not only does it emphasize that some sums are more likely than others, but the calculations specifically enumerate how much more likely some sums are than others. For example, the chance of observing a sum of 5 is six times that of observing a sum of 3. While these calculations are quite routine using modern methods for counting configurations and orderings, it is notable that De Vetula is the first known exposition of this type of problem in probability.