498 CHAPTER 8 Discrete Probability
The probability P(E) of E is
P(E) = p(to)
WEE
= (0.9)(0.8)(0.5) + (0.9)(0.8)(0.5) + (0.9)(0.2)(0.5) + (0.1)(0.8)(0.5)
+(0.1)(0.2)(0.5)
= 0.36 + 0.36 + 0.09 + 0.04 + 0.01=0.86 M
8.3.3 Bernoulli Trial Processes
An important special case of a cross product sample space occurs when an experiment with
just two outcomes is repeated numerous times. Typically, the number n of repetitions can
be controlled, and we want to predict what happens as a function of n.
Definition 3. A Bernoulli trial process (abbreviated Bernoulli process) is a sequence of
repetitions, called trials, of an experiment with a two-element sample space. It is assumed
that the trials have no influence on one another. The two possible outcomes of a trial need
not be equally likely.
Bernoulli processes arise in many contexts. Flipping a coin over and over can be re-
garded as a Bernoulli process. Another example is the sending of binary digits, or bits,
over a communication line.
Often, the two elements of the sample space Q2 of a Bernoulli trial are labeled success
and failure. For example, when we flip a coin, we can label the outcome heads a success
and the outcome tails a failure. Similarly, if a bit remains unchanged during transmission
over a communication line, we say that a success has occurred; if the bit changes, we say
that a failure has occurred.
In the context of a Bernoulli process, it is common to denote the probability of the
success outcome with the letter p and the probability of the failure outcome with the letter
q. Since the sample space of a Bernoulli trial has only two elements, p = 1 - q where0< p, q < 1.
Notation. Since the probability density function on an arbitrary sample space is often
denoted by the letter p, keep in mind that the letter p can have two usages. For example,
suppose that Q2 = {Wl, (021 is a two-element sample space where w~l means success and (02
means failure. If we say that the probability of success is p and the probability of failure
is q, then we are really defining a probability density p on Q given by p(w0l) = p and
p(oJ2) = q.
A Bernoulli process can be modeled by a cross product sample space as follows. Sup-
pose the trials of the process have outcomes in the two-element sample space Q. To regard
a sequence of n trials as just one experiment, we form the cross product sample space
consisting of n terms:
Qn • X ... X Q2The probability density function associated with the cross product (see Definition 2) mod-
els the assumption that the trials do not influence one another.