7.3 Binomial Distribution and Binomial Experiments
7.3 Binomial Distribution and Binomial Experi-
ments
Learning Objectives
- Apply techniques to estimate the probability of a population proportion of outcomes in a survey and experi-
ment. - Understand the conditions needed to conduct a binomial experiment.
Introduction
A probability distribution shows the possible numerical outcomes of a chance process and from this the probability of
any set of possible outcomes can be deduced. As seen in previous lessons, for many events probability distribution
can be modeled by the normal curve. One type of probability distribution that is worth examining is a binomial
distribution.
In probability theory and statistics, thebinomial distributionis the discrete probability distribution of the number
of successes in a sequence of “n” independent yes/no experiments, each of which yields success with probability
“p” (such experiments are called Bernoulli experiments).
To conduct abinomial experimenta random sample (nindependent trials) must be chosen, and the number
of successes(x)determined. Then the sample proportion ˆpcan be found to predict the population proportion
(probability of success, p).Many experiments involving random variables are simply exercises in counting the
number of successes innindependent trials, such as
- The number of people with type O blood in a random sample of 10 people (a person either has type O blood
or doesn’t.) - The number of doubles in eight rolls of a pair of dice (doubles either show up on each roll or they don’t.)
- The number of defective light bulbs in a sample of 30 bulbs (either the bulb is defective or it isn’t.)
Binomial Experiments
These events are called binomial because each one has two possible outcomes. Let’s examine an actual binomial
situation. Suppose we present four people with two cups of coffee (one percolated and one instant) to discover the
answer to this question: “If we ask four people which is percolated coffee and none of them can tell the percolated
coffee from the instant coffee, what is the probability that two of the four will guess correctly?” We will present
each of four people with percolated and instant coffee and ask them to identify the percolated coffee. The outcomes
will be recorded by usingCfor correctly identifying the percolated coffee andIfor incorrectly identifying it. The
following list of 16 possible outcomes, all of which are equally likely if none of the four can tell the difference and
are merely guessing, is shown below: