Chapter 1: Basic Concepts

Also, when the probability of an event is close to zero, the occurrence of the event is relatively unlikely. For example, if the chances that you will win a certain lottery are 0.001 or 1 in 1000, you probably won’t win, unless of course, you are very “lucky.” When the probability of an event is 0.5 or

, there is a 50-50 chance that the event will happen—the same probability of getting one of the two outcomes when flipping a coin. When the probability of an event is close to one, the event is almost certain to occur. For example, if the chance of it snowing tomorrow is 90%, more than likely you’ll see some snow. See Figure 1-1.

PROBABILITY SIDELIGHT: Brief History of Probability

The concepts of probability are as old as humans. Paintings in tombs excavated in Egypt showed that people played games based on chance as early as 1800 BCE. One game was called “Hounds and Jackals” and is similar to the present-day game of “Snakes and Ladders.”

Ancient Greeks and Romans made crude dice from various items such as animal bones, stones, and ivory. When some of these items were tested recently, they were found to be quite accurate. These crude dice were also used in fortune telling and divination.

Emperor Claudius (10 BCE–54 CE) is said to have written a book entitled How To Win at Dice. He liked to play so much that he had a special dice board in his carriage.

No formal study of probability was done until the 16th century when Girolamo Cardano (1501–1576) wrote a book on probability entitled The Book on Chance and Games. Cardano was a philosopher, astrologer, physician, mathematician, and gambler. In his book, he also included techniques on how to cheat and how to catch others who are cheating. He is believed to be the first mathematician to formulate a definition of classical probability.

During the mid-1600s, a professional gambler named Chevalier de Mere made a considerable amount of money on a gambling game. He would bet unsuspecting patrons that in four rolls of a die, he could obtain at least one 6. He was so successful at winning that word got around, and people refused to play. He decided to invent a new game in order to keep winning. He would bet patrons that if he rolled two dice 24 times, he would get at least one double 6. However, to his dismay, he began to lose more often than he would win.

Unable to figure out why he was losing, he asked a renowned mathematician, Blaise Pascal (1623–1662) to study the game. Pascal was a child prodigy when it came to mathematics. At the age of 14, he participated in weekly meetings of the mathematicians of the French Academy. At the age of 16, he invented a mechanical adding machine.

Because of the dice problem, Pascal became interested in studying probability and began a correspondence with a French government official and fellow mathematician, Pierre de Fermat (1601–1665). Together the two were able to solve de Mere’s dilemma and formulate the beginnings of probability theory.

In 1657, a Dutch mathematician named Christian Huygens wrote a treatise on the Pascal–Fermat correspondence and introduced the idea of mathematical expectation. (See Chapter 5.)

Abraham de Moivre (1667–1754) wrote a book on probability entitled Doctrine of Chances in 1718. He published a second edition in 1738.

Pierre Simon Laplace (1749–1827) wrote a book and a series of supplements on probability from 1812 to 1825. His purpose was to acquaint readers with the theory of probability and its applications using everyday language. He also stated that the probability that the sun will rise tomorrow is

Simeon-Denis Poisson (1781–1840) developed the concept of the Poisson distribution. (See Chapter 8.)

Also during the 1800s, a mathematician named Carl Friedrich Gauss (1777–1855) developed the concepts of the normal distribution. Earlier work on the normal distribution was also done by DeMoivre and Laplace, unknown to Gauss. (See Chapter 9.)

In 1895, the Fey Manufacturing Company of San Francisco invented the first automatic slot machine. These machines consisted of three wheels that were spun when a handle on the side of the machine was pulled. Each wheel contained 20 symbols; however, the number of the same symbols was not the same on each wheel. For example, the first wheel may have 6 oranges, while the second wheel has 3 oranges, and the third wheel has 1 orange. When a person gets two oranges, the person may think that he has almost won by getting 2 out of 3 equitable symbols when the real probability of winning is much smaller.

In the late 1940s, two mathematicians, John von Neumann and Stanislaw Ulam, used a computer to simulate probability experiments. This method is called the Monte Carlo method. (See Chapter 10.)

Today probability theory is used in insurance, gambling, war gaming, the stock market, weather predicting, and many other areas.

Probability is the mathematics of chance. There are three types of probability: classical probability, empirical probability, and subjective probability. Classical probability uses sample spaces. A sample space is the set of outcomes of a probability experiment. The range of probability is from 0 to 1. If an event cannot occur, its probability is 0. If an event is certain to occur, its probability is 1. Classical probability is defined as the number of ways (outcomes) the event can occur divided by the total number of outcomes in the sample space.

Experiment	Sample Space
Toss one coin	H, T*
Roll a die	1, 2, 3, 4, 5, 6
Toss two coins	HH, HT, TH, TT
*H = heads; T = tails

Rank	Frequency
Freshmen	6
Sophomores	2
Juniors	10
Seniors	12
TOTAL	30

Age	Number
20–29	10
30–39	36
40–49	22
50–59	18
60 or older	14
	100

CHAPTER OBJECTIVES

QUIZ