11.1 Empirical and Theoretical Probabilities 657 How do researchers determine if a new potential drug is superior to an existing drug? How do automobile manufacturers determine the chance that a particular auto part will fail before the warranty expires? Which casino games provide the greatest chance of winning? These and similar questions can be answered once you have an understanding of probability. Empirical and Theoretical Probabilities SECTION 11.1 LEARNING GOALS Upon completion of this section, you will be able to: ■ Understand the history and nature of probability. ■ Understand empirical probability and the law of large numbers. ■ Understand theoretical probability. Why This Is Important Understanding probability is essential to our daily lives. Probability is used to determine insurance premiums, the length of a warranty, and the expected gain or loss of business ventures. Probability is also important in medical research as well as many other areas. In this section we introduce some of the basic concepts of probability including empirical and theoretical probability. We also discuss the law of large numbers and how probability was used in the foundation of genetics. We begin with the history of probability. Profile in Mathematics Jacob Bernoulli (1654–1705) Swiss mathematician Jacob Bernoulli was considered a pioneer of probability theory. He was part of a famous family of mathematicians and scientists. In Ars Conjectandi, published posthumously in 1713, he proposed that an increased degree of accuracy can be obtained by increasing the number of trials of an experiment. This theorem, called Bernoulli’s theorem (of probability), is also known as the law of large numbers. Bernoulli’s theorem of fluid dynamics, used in aircraft wing design, was developed by Daniel Bernoulli (1700–1782). Daniel was one of Jacob’s younger brother Johann’s three sons. The History and Nature of Probability The study of probability originated from the study of games of chance. Mathematical problems relating to games of chance were studied by a number of mathematicians of the Renaissance. Italy’s Girolamo Cardano (1501–1576) in his Liber de Ludo Aleae (book on the games of chance) presents one of the first systematic computations of probabilities. Although it is basically a gambler’s manual, many consider it the first book ever written on probability. A short time later, two French mathematicians, Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665), worked together studying “the geometry of the die.” In 1657, Dutch mathematician Christian Huygens (1629–1695) published De Ratiociniis in Luno Aleae (on ratiocination in dice games), which contained the first documented reference to the concept of mathematical expectation (see Section 11.3). Swiss mathematician Jacob Bernoulli (1654–1705), whom many consider the founder of probability theory, is said to have fused pure mathematics with the empirical methods used in statistical experiments. The works of Pierre-Simon de Laplace (1749–1827) dominated probability throughout the nineteenth century. Today, probability is involved in many areas of mathematics and science and is used in industrial and business applications. Next, we introduce some of the definitions essential to our understanding of the nature of probability. Definition: Experiment An experiment is a controlled operation that yields a set of results. Some of the experiments we will study in this section include tossing a coin, rolling a single die, and drawing a card from a standard 52-card deck of playing cards. Another example of an experiment is a medical researcher administering an experimental drug to patients to determine their reactions. Gorodenkoff/ Shutterstock
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