Probability

CHAPTER 7 Probability




The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk: the notion that the future is more than a whim of the gods and that men and women are not passive before nature.


—Peter L. Bernstein1


The science of inferential statistics did not develop overnight. It evolved over hundreds of years from basic mathematical principles into a discipline that uses observations to predict the future. The probability theories upon which inferential statistics is built were revealed sequentially through a diligent process that reflects several centuries of intellectual curiosity and pursuit.



THE HISTORY OF PROBABILITY THEORY


Statistics is a relatively recent product of the last few centuries. The earliest counting systems were able to keep track of large numbers of items, but did not allow for the abstract idea of “none” or for negative numbers. The modern arithmetic that we use today had its humble origin about 1500 years ago, when the Hindus developed a numbering system that incorporated the concept of zero. This notion allowed for the development of a more standard approach to counting in which a limited number of digits (0 through 9) could be placed side-by-side, and their relative value depended on the column (1s, 10s, 100s, etc.) they occupied. Now there was no limit on the value that a number could express.


The Arabs adopted the new system when they traveled throughout India, and developed it further. In the ninth century an Arabic mathematician named Al-Khowarizmi wrote one of the earliest mathematical works on simple algebraic equations. His name ultimately evolved into the word algorithm, which means “rules for computing.” Another of his works, Hisab al-jabr w’ almuqabalah, which means “science of transposition and cancellation,” gives us the word for algebra (al-jabr). The new numbers were met with some resistance in Western Europe, but over the next few centuries they were eventually adopted as a logical way of expressing values and solving equations. This novel system allowed for the development of probability theory, which is based on equations that enable us to quantify risk.


Humans like the comfort of predictable outcomes, but we live in an imperfect world. We strive to make failure-free devices, but any equipment we produce will have a failure rate. We cannot say with confidence that a bad event will not happen when we travel. We also cannot guarantee that a recommended treatment will work. We have no choice but to accept some uncertainty about the future. However, guided by the laws of probability, we can make informed choices that allow us to maximize the chance of a good outcome.


The radical concept that humans could predict outcomes is relatively new. Until the past few centuries, people believed that future events were controlled by an omnipotent deity. They were powerless to mold their destiny. Without the framework of the science of logic, they relied on the alluring predictions of mystics and soothsayers. As laws of physics were discovered during the Middle Ages, people began to realize that there was some order to the events of the heavens and the physical world. Only then did the aura of mysticism surrounding secular events began to dissolve.


People have always been intrigued by games of chance. Evidence of gambling has been found on Egyptian paintings that date back as early as 3500 B.C., and accounts of this pastime have been recorded throughout history in all societies and economic classes. As people began to appreciate that their world responded to the laws of physics, they also discovered that certain games had outcomes which could be predicted by mathematical formulas. The desire to predict these outcomes led to an interest in the “science of choice.”


Several great works were written during the Renaissance on the topic of chance. One of the most notable was a treatise by a sixteenth-century Italian physician named Girolamo Cardano. He was an avid gambler who developed the algebraic formulas to calculate the expected outcomes when throwing dice. These formulas are used today by casino owners to ensure that they have the advantage in the long run and do not lose money.


The outcomes in games of chance (such as dice) can be calculated mathematically because the chance of each occurrence is known. In a fair game of dice, we know the probability of getting a sum of 7 when two dice are rolled. There are six combinations that could produce a 7 out of a total of 36 possible combinations. The probability formulas allow us to calculate the frequency at which we can expect to roll a 7 when we throw the dice more than once. An analogous situation is the flip of a coin. We know the probability of getting heads is 50% but, using the binomial formulas, we can also calculate the chances of getting heads one-quarter of the time over multiple coin flips.


A century before Cardano’s formulas were presented, an Italian monk named Luca Pacioli had posed the question of how the stakes should be divided in an unfinished game when one of the players is ahead. This riddle remained unsolved for two centuries, until 1654, when the famous French mathematicians Pascal and Fermat were able to solve the puzzle by using formulas to predict what would most likely happen if the game kept going.


Predicting chance occurrences in events such as coin flips and rolls of the dice led to more hypothetical inquiries. Is there a way to predict events in which the probability of a particular outcome is unknown? Jacob Bernoulli was a mathematician who was intrigued by the idea of applying probability theory to these abstract situations. In 1713 his essay Ars Conjectandi (The Art of Conjecture

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Jun 18, 2016 | Posted by in BIOCHEMISTRY | Comments Off on Probability

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