Page 475 - Discrete Mathematics and Its Applications

454 7 / Discrete Probability

Remark: We will not discuss probabilities of events when the set of outcomes is not finite or
countable, such as when the outcome of an experiment can be any real number. In such cases,
integral calculus is usually required for the study of the probabilities of events.

We can model experiments in which outcomes are either equally likely or not equally likely
by choosing the appropriate function p(s), as Example 1 illustrates.

EXAMPLE 1 What probabilities should we assign to the outcomes H (heads) and T (tails) when a fair coin
is flipped? What probabilities should be assigned to these outcomes when the coin is biased so
that heads comes up twice as often as tails?

Solution: For a fair coin, the probability that heads comes up when the coin is flipped equals
the probability that tails comes up, so the outcomes are equally likely. Consequently, we assign
the probability 1/2 to each of the two possible outcomes, that is, p(H) = p(T ) = 1/2.
For the biased coin we have

p(H) = 2p(T ).

Because

p(H) + p(T ) = 1,

it follows that

2p(T ) + p(T ) = 3p(T ) = 1.

We conclude that p(T ) = 1/3 and p(H) = 2/3. ▲

DEFINITION 1 Suppose that S is a set with n elements. The uniform distribution assigns the probability 1/n
to each element of S.

We now define the probability of an event as the sum of the probabilities of the outcomes
in this event.

DEFINITION 2 The probability of the event E is the sum of the probabilities of the outcomes in E. That is,

p(E) = p(s).
s∈E

(Note that when E is an infinite set, p(s) is a convergent infinite series.)
s∈E
Note that when there are n outcomes in the event E, that is, if E ={a 1 ,a 2 ,...,a n }, then
n

p(E) = i=1 p(a i ). Note also that the uniform distribution assigns the same probability to
an event that Laplace’s original definition of probability assigns to this event. The experiment
of selecting an element from a sample space with a uniform distribution is called selecting an
element of S at random.

EXAMPLE 2 Suppose that a die is biased (or loaded) so that 3 appears twice as often as each other number
but that the other five outcomes are equally likely. What is the probability that an odd number
appears when we roll this die?