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Chapter 8: Discrete Probability Distributions
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Definition: Bernoulli Trials
Bernoulli trials occur when a Bernoulli experiment is repeated several
independent times so that the probability of success, say
, remains the same
from trial to trial.
In a sequence of Bernoulli trials, we are often interested in the total number
of successes. For example, if we examine 25 items off a production line, we
may be interested in the number of defectives we observe in the sample of
size 25. If we let
X
be the random variable that represents the number of
defective items in the sample, then
X
is called a
binomial random variable
and the associated experiment, a
binomial experiment
.
A
binomial experiment
is a random experiment which satisfies the
following five conditions:
1. There are two possible outcomes (success or failure) on each trial.
2. There are a fixed number of trials, say
.
3. The probability of success, say
, is fixed from trial to trial.
4. The trials are independent.
5. The binomial random variable is the number of successes in the
trials.
Binomial experiments occur quite frequently in the real world, and a model
has been developed to help compute the probabilities associated with such
experiments. Before we discuss the binomial distribution, however, we need
to introduce the concept of factorials.
Factorials
Suppose there are five job positions to be filled by five different applicants.
There will be 5 choices for the first position. Once this is filled, there are
only four applicants remaining; thus there will be 4 choices for the second
position. We can continue in this manner until all of the positions are filled.
Note that there will only be one choice for the last position. By a counting
principle, there will be 5
4
3
2
1 = 120 ways of filling the five positions.
We say that there are 5
factorial ways
of filling these positions.
Notation
:
We will use the symbol “!”
to represent factorials.