Page 474 - Discrete Mathematics and Its Applications

7.2 Probability Theory 453

similar questions, we will introduce the concept of conditional probability. Does knowing that
the first flip comes up heads change the probability that heads comes up three times? If not,
these two events are called independent, a concept studied later in this section.
Many questions address a particular numerical value associated with the outcome of
an experiment. For instance, when we flip a coin 100 times, what is the probability that
exactly 40 heads appear? How many heads should we expect to appear? In this section we
will introduce random variables, which are functions that associate numerical values to the
outcomes of experiments.

Assigning Probabilities

Let S be the sample space of an experiment with a finite or countable number of outcomes. We
assign a probability p(s) to each outcome s. We require that two conditions be met:

(i) 0 ≤ p(s) ≤ 1 for each s ∈ S

and

(ii) p(s) = 1.
s∈S
Condition (i) states that the probability of each outcome is a nonnegative real number no greater
than 1. Condition (ii) states that the sum of the probabilities of all possible outcomes should
be 1; that is, when we do the experiment, it is a certainty that one of these outcomes occurs.

(Note that when the sample space is infinite, s∈S p(s) is a convergent infinite series.) This is
a generalization of Laplace’s definition in which each of n outcomes is assigned a probability
of 1/n. Indeed, conditions (i) and (ii) are met when Laplace’s definition of probabilities of
equally likely outcomes is used and S is finite. (See Exercise 4.)
Note that when there are n possible outcomes, x 1 ,x 2 ,...,x n , the two conditions to be met
are

(i) 0 ≤ p(x i ) ≤ 1 for i = 1, 2,...,n

and

n

(ii) p(x i ) = 1.
i=1
The function p from the set of all outcomes of the sample space S is called a probability
distribution.
To model an experiment, the probability p(s) assigned to an outcome s should equal the limit
of the number of times s occurs divided by the number of times the experiment is performed,
as this number grows without bound. (We will assume that all experiments discussed have
outcomes that are predictable on the average, so that this limit exists. We also assume that the
outcomes of successive trials of an experiment do not depend on past results.)

HISTORICAL NOTE The Chevalier de Méré was a French nobleman, a famous gambler, and a bon vivant.
He was successful at making bets with odds slightly greater than 1/2 (such as having at least one six come
up in four tosses of a fair die). His correspondence with Pascal asking about the probability of having at least
one double six come up when a pair of dice is rolled 24 times led to the development of probability theory.
According to one account, Pascal wrote to Fermat about the Chevalier saying something like “He’s a good guy
but, alas, he’s no mathematician.”