Bernoulli II

Part One.
Here Bernoulli sets forth some basic principles, taken from Huygen’s Treatise on Reckoning and Games of Chance on which he builds in the later sections, in fourteen Propositions followed by application in five Problems together with solutions. Propositions 9-14 concern the casting of lots with dice and are annotated by Bernoulli.
“ON DICE”
“The following questions can be asked about dice. With one die, in how many attempts might one undertake to throw a six, or any on of the other numbers of points? Similarly, with two dice, in how many attempts might one undertake to throw two sixes? Or with three dice, three sixes? There are many similar questions.
“To answer these questions consider the following. First, with one die there are six different throws, all of which can happen equally easily. (I assume that the die has the shape of a perfect cube.) Next, with two dice there are 36 different throws, which again can happen equally easily. This is because with each throw of the one die any one of the six throws of the other die can happen, and six times six equals 36. Similarly, there are 216 different throws with three dice, because any of the 36 throws of the two dice can happen together with each of the six throws of the third, and six times 36 makes 216 throws. In the same way it is clear that with four dice there are six times 216, or 1,296 throws. By continuing we can calculate the throws for any number of dice; every time we add another die we multiply the preceding number of throws by six.
“Next, note that there is just one way to get a 2 or12 with two dice, but two ways to get a 3 or 11. Indeed, if we call the dice A and B, then we can get 3 points either from 1 point on A and 2 on B or from 2 on A and 1 on B. Similarly, we can get 11 points either from 5 points on A and 6 on B or from 6 on A and 5 on B. There are three ways of throwing a 4, namely 1 on A and 3 on B, 3 on A and 1 on B, or 2 on A and 2 on B.
Similarly, there are three ways of throwing 10.
There are four ways of throwing 5 or 9.
There are five ways of throwing 6 or 8.
There are six ways of throwing 7.”
“With three dice to throw points, there are different throws.”
(Here, the “points” Bernoulli refers to are simply the sum of the dots on one or both dice.)
At this point, there is a note showing how, in the same manner, one can find the number of ways to throw any number of points with four or more dice. This proves to be very lengthy and difficult to keep track of the Modes (or variations), and so a table is given to show a more simple way of getting the same outcome. This table is easy to construct if you follow the directions carefully. It can be used for any number of dice, but is only constructed for up to six in the books table and in the reconstruction on the following page. The table is used by first choosing the Number of Dice you are using, (represented by the first set of Roman numerals on the top left) and following that row down to the number of points you wish to roll. Then, follow down the column to the section labeled Number of Throws for Given Number of Dice and find the section with Roman numerals equal to your chosen Number of Dice. This is the number of different rolls that can yield your chosen number of points. For example, let us say you have four dice and you wish to roll a total of “4.” You can find the number of possible ways this can occur by starting at the top of the page and following across the row with the Roman numeral IV over to the number 4. (Here it is the first number that occurs.) Now, follow the column down through three sections until you reach section IV. The number in the top row of this section, which lines up with “4” is the number of ways you can roll a “4.” Here the chart shows there is only 1 possible way to roll a “4” with four dice. The same is true if you wish to roll a total of 24 points with four dice – there is only 1 way. Follow the same protocol to find that, using four dice there are 4 throws that can add up to 5 or 23, ten different ways you can roll a 6 or 22, and twenty rolls that can yield a 7 or 21. Using a table such as this can help you keep track of your odds for rolling certain numbers and, consequently, how much you can safely bet on each roll.
This leads directly in to Part Two: