Introduction to Probability and Statistics for Engineers and Scientists

62 Chapter 3:Elements of Probability

Now it follows from Axioms 2 and 3 that

1 =P(S)=P({ 1 })+ ··· +P({N})=Np

which shows that

P({i})=p=1/N

From this it follows from Axiom 3 that for any eventE,

P(E)=

Number of points inE

N

In words, if we assume that each outcome of an experiment is equally likely to occur, then
the probability of any eventEequals the proportion of points in the sample space that are
contained inE.
Thus, to compute probabilities it is often necessary to be able to effectively count the
number of different ways that a given event can occur. To do this, we will make use of the
following rule.

BASIC PRINCIPLE OF COUNTING

Suppose that two experiments are to be performed. Then if experiment 1 can result in
any one ofmpossible outcomes and if, for each outcome of experiment 1, there aren
possible outcomes of experiment 2, then together there aremnpossible outcomes of the
two experiments.

Proof of the Basic Principle

The basic principle can be proven by enumerating all the possible outcomes of the two
experiments as follows:

(1, 1), (1, 2),...,(1,n)

(2, 1), (2, 2),...,(2,n)

..

.

(m, 1), (m, 2),...,(m,n)

where we say that the outcome is (i,j) if experiment 1 results in itsith possible outcome
and experiment 2 then results in thejth of its possible outcomes. Hence, the set of
possible outcomes consists ofmrows, each row containingnelements, which proves the
result. ■

EXAMPLE 3.5a Two balls are “randomly drawn” from a bowl containing 6 white and 5
black balls. What is the probability that one of the drawn balls is white and the other black?