Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

7.2 History

ROLL DEM' BONES

• The mathematical study of probability probably was delayed for centuries because of mysticism.

Throughout the ages, people have responded to the problem of what to do about the future in different ways. For many ancient societies, the unknown future was considered to be the province of the gods. Understanding and making predictions about this future was left to religious figures and oracles. These people employed a number of methods and devices with which they supposedly divined the will of the gods.

Some of the most common tools of the ancient religious diviner were astragali. These were bones, taken from the ankles of sheep, that would be cast and interpreted. Astragali commonly had six sides, but they were very asymmetrical. Often they were cast in groups, with the specific combinations of values revealing the name of the god who could be expected to affect the future affairs of the people. For example, if the bones said that Zeus was at work, there would be reason for hope. If the bones said that Cronos was in charge, then the people knew to prepare for the worst.

Item 2242 / Kathleen Cohen, KNUCKLE-BONES (2008). Courtesy of Kathleen Cohen.

Item 2241 / Kathleen Cohen, DICE; CHECKERS; AND ASTRAGALUS (FOR KNUCKLE-BONES). (2008). Courtesy of Kathleen Cohen.

Item 2240 / Kathleen Cohen, DICE (2008). Courtesy of Kathleen Cohen.

Gradually, technology enabled the development of more-regularly-shaped "prediction" devices. The first dice, made of pottery, are thought to have appeared in ancient Egypt. By the time of the flowering of Greek culture, dice were quite common, both for fortune-telling and for gaming. Dice have always been popular tools in recreational gaming, or gambling, precisely because they are thought to be random-event generators. "The roll of the dice" is thought to be the ultimate unknown, so dice are thought to be somewhat fair arbiters. This assumes of course that the dice are perfectly symmetrical and evenly weighted, which early dice often were not. Discoveries of ancient loaded dice reveal that, even though ancient people did not have a mathematical understanding of probability, they knew how to weight games in their favor.

One might think that the Greeks, who embraced a central role for mathematics in the world of the mind, would have discovered the features of probabilistic thinking. Evidence shows that they did not. It is thought that the Greeks deemed matters of chance to be the explicit purview of the gods. According to this view, they believed that any attempt to understand what happens and what should happen was a trespass into the territory of the gods. It was not of human concern.

Additionally, the Greeks favored understanding based on logical reasoning over understanding based on empirical observations. One of the concepts at the heart of our modern understanding of probability is concerned with how actual results compare with theoretical predictions. This type of empirical thinking often took a back seat to logical axiomatic arguments in the mathematics of ancient Greece.

The mathematics of probability went undiscovered for centuries, but gambling, especially with dice, flourished. It seems that dice, in some form or another, have been a constant feature of civilization from the time of the Greeks onward. The Romans were fond of them, as were the knights of the Middle Ages, who played a game called Hazard, an early forerunner of the modern game of craps, thought to have been brought back from the Crusades.

A RATHER LATE DEVELOPMENT

• Renaissance mathematicians took the first strides toward understanding chance in an abstract way.
• Pascal's and Fermat's solutions to the "Problem of the Points" provided an early glimpse of how to use mathematics to say definite things about unknown future events.
• Tree diagrams are useful for keeping track of possible outcomes.

It was not until the Renaissance that fascination with dice as an instrument of gambling led to the first recorded abstract ideas about probability. The man most responsible for this new way of thinking was a quintessential Renaissance man, an accomplished doctor and mathematician by the name of Girolamo Cardano.

Cardano was famous in the mathematical world for many things, most notably his general solutions to cubic equations. As a doctor, he was among the best of his day. His passion, however, was to be found at the dice table. He was a fanatic and compulsive gambler, once selling all of his wife's possessions for gambling stakes. Out of his obsession grew an interest in understanding analytically, and, thus, mathematically, the odds of rolling certain numbers with dice. In particular, he figured out how to express the chances of something happening as a ratio of the number of ways in which the event could happen to the total number of outcomes.

For example, what's the probability of rolling a 4 with two regular dice? There are thirty-six possible equally likely outcomes when a pair of dice is rolled, and of these only three combinations (1 and 3, 2 and 2, and 3 and 1) produce a total value of 4. So, the probability of rolling a 4 is or . This seemingly straightforward observation was the first step toward a robust mathematical understanding of the laws of chance. Dr. Cardano penned his thoughts on the matter around 1525, but his discoveries were to go unpublished until 1663. By that time, two Frenchmen had already made significant progress of their own.

In the mid-1600s, the Chevalier de Méré, a wealthy French nobleman and avid gambler, wrote a letter to one of the most prominent French mathematicians of the day, Blaise Pascal. In his letter to Pascal, he asked how to divide the stakes of an unfinished game. This so-called "problem of the points" was framed as follows:

Suppose that two men are playing a game in which the first to win six points takes all the money. How should the stakes be divided if the game is interrupted when one man has five points and the other three?

Pascal consulted with Pierre de Fermat, another very prominent mathematician of the day, in a series of letters that would become the basis for much of modern probability theory. Fermat and Pascal approached the problem in different ways. Fermat tended to use algebraic methods, while Pascal favored geometric arguments. Both were concerned basically with counting. They figured that in order to divide the stakes properly, they could not simply divide them in half, because that would be unfair to the man who was in the lead at the time of the game's cessation. A proper division of the stakes would be based on how likely it was that each player would have won had the game continued to completion. The player in the lead could win in one more round. The trailing player would need at least three more rounds to win. Furthermore, he must win in each of those three rounds. Therefore, the two Frenchmen reasoned, the proper division of the stakes should be based on how likely it was that the trailing player would win three rounds in a row.

The trailing player has a one-in-two chance of winning the next round. Provided he wins, he then has another one-in-two chance of winning the following round. So, after two rounds, there are four possible outcomes, only one of which is favorable to the trailing player. If he should happen to win the first two rounds, he then again has a one-in-two chance of winning the third round. Let's take a look at how all of this information can be represented in a tree diagram.

As we see in the tree diagram above, only one of the eight possible outcomes results in the trailing player winning the stakes. Therefore, the trailing player should be awarded of the pot, with the remaining going to the player who was winning at the time of the interruption. This method of enumerating and examining the possible outcomes of random events was a crucial link in the mathematical conquest of the unpredictable.