Page 467 - Discrete Mathematics and Its Applications

446 7 / Discrete Probability

some key concepts in probability theory, including conditional probability, independence of
events, and random variables. In Section 7.4 we will introduce the concepts of the expectation
and variance of a random variable.

Finite Probability

An experiment is a procedure that yields one of a given set of possible outcomes. The sample
space of the experiment is the set of possible outcomes. An event is a subset of the sample
space. Laplace’s definition of the probability of an event with finitely many possible outcomes
will now be stated.

DEFINITION 1 If S is a finite nonempty sample space of equally likely outcomes, and E is an event, that

|E|
is, a subset of S, then the probability of E is p(E) = .
|S|

The probability of an According to Laplace’s definition, the probability of an event is between 0 and 1. To see this,
event can never be note that if E is an event from a finite sample space S, then 0 ≤|E|≤|S|, because E ⊆ S.
negative or more than
one! Thus, 0 ≤ p(E) =|E|/|S|≤ 1.
Examples 1–7 illustrate how the probability of an event is found.
EXAMPLE 1 An urn contains four blue balls and five red balls. What is the probability that a ball chosen at
random from the urn is blue?

Solution: To calculate the probability, note that there are nine possible outcomes, and four of
these possible outcomes produce a blue ball. Hence, the probability that a blue ball is chosen
is 4/9. ▲

EXAMPLE 2 What is the probability that when two dice are rolled, the sum of the numbers on the two dice
is 7?

Solution: There are a total of 36 equally likely possible outcomes when two dice are rolled.
(The product rule can be used to see this; because each die has six possible outcomes, the total

GIROLAMO CARDANO (1501–1576) Cardano , born in Pavia, Italy, was the illegitimate child of Fazio
Cardano, a lawyer, mathematician, and friend of Leonardo da Vinci, and Chiara Micheria, a young widow.
In spite of illness and poverty, Cardano was able to study at the universities of Pavia and Padua, from where
he received his medical degree. Cardano was not accepted into Milan’s College of Physicians because of his
illegitimate birth, as well as his eccentricity and confrontational style. Nevertheless, his medical skills were
highly regarded. One of his main accomplishments as a physician is the first description of typhoid fever.
Cardano published more than 100 books on a diverse range of subjects, including medicine, the natural
sciences,mathematics,gambling,physicalinventionsandexperiments,andastrology.Healsowroteafascinating
autobiography. In mathematics, Cardano’s book Ars Magna, published in 1545, established the foundations of
abstract algebra. This was the most comprehensive book on abstract algebra for more than a century; it presents many novel ideas of
Cardano and of others, including methods for solving cubic and quartic equations from their coefficients. Cardano also made several
important contributions to cryptography. Cardano was an advocate of education for the deaf, believing, unlike his contemporaries,
that deaf people could learn to read and write before learning to speak, and could use their minds just as well as hearing people.
Cardano was often short of money. However, he kept himself solvent through gambling and winning money by beating others
at chess. His book about games of chance, Liber de Ludo Aleae, written in 1526 (but published in 1663), offers the first systematic
treatment of probability; it also describes effective ways to cheat. Cardano was considered to be a man of dubious moral character;
he was often described as a liar, gambler, lecher, and heretic.