Math 141: Business Mathematics I
Fall 2015
§8.4 The Binomial Distribution
Instructor: Yeong-Chyuan Chung
Some experiments have (or may be viewed as having) only two possible outcomes. Some
examples of such experiments are:
• Tossing a coin and observing whether we get heads or tails.
• Picking a card from a deck of cards and observing whether we get a heart.
• Picking a marble from a container of marbles of various colors and observing whether
we get a marble of a particular color.
Experiments with only two possible outcomes are called Bernoulli trials, or binomial
trials. We typically refer to one of the outcomes as a success, and to the other outcome as a
failure. A sequence of such trials is called a binomial experiment. A binomial experiment
has the following properties:
1. The number of trials in the experiment is fixed.
2. There are exactly two possible outcomes in each trial: “success” and “failure.”
3. The probability of success in each trial is the same.
4. The trials are independent of each other.
Example (Exercise 2,4,6 in the text). Determine whether each of the following experiments
is a binomial experiment.
(a) Rolling a fair die and observing the number of times the die is thrown until a 6 appears
uppermost.
(b) A card is selected from a deck of 52 cards, and its color is observed. A second card is
then drawn (without replacement), and its color is observed.
(c) Recording the number of hits a baseball player, whose batting average is .325, gets after
being up to bat five times.
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§8.4 The Binomial Distribution
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In a binomial experiment, we usually use the following notation:
• The number of trials is denoted by n.
• The probability of success in a trial is denoted by p (and so the probability of failure
is 1 − p).
With this notation, the probability of exactly r successes in n independent trials is given by
C(n, r)pr (1 − p)n−r .
In this formula, the term pr corresponds to the r successes, the term (1 − p)n−r corresponds
to the n − r failures, and the term C(n, r) accounts for the number of ways to get exactly r
successes in n trials.
A random variable X that gives the number of successes in a binomial experiment is called
a binomial random variable, and the probability distribution of X is called a binomial
distribution. Since P (X = r) means “the probability of exactly r successes”, we may write
P (X = r) = C(n, r)pr (1 − p)n−r .
Using the graphing calculator to compute probabilities in binomial
experiments
There are two commands in the calculator that can be used to compute probabilities in
binomial experiments - binompdf and binomcdf. Both can be found by pressing 2nd
VARS.
binompdf is used to compute the probability of exactly r successes in n trials.
We just need to type binompdf(n,p,r) where p is the probability of success.
binomcdf is used to compute the probability of at most r successes in n trials.
The “c” stands for “cumulative”, and we just need to type binomcdf(n,p,r).
Example (Exercise 12 in the text). Find the probability of exactly three successes in six
trials of a binomial experiment in which p = 21 .
§8.4 The Binomial Distribution
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Example (Exercise 16 in the text). Find the probability of at least one failure in five trials
of a binomial experiment in which p = 13 .
Example (Exercise 40 in the text). A study conducted in 2012 determined that 23% of the
population has normal blood pressure (with medication). What is the probability that at most
3 out of 10 people chosen at random have normal blood pressure (with medication)?
Example (Exercise 46 in the text). The probability that an airplane engine will fail in a
transcontinental flight is .001. Assuming that engine failures are independent of each other,
what is the probability that on a certain transcontinental flight, a four-engine plane will
experience:
(a) Exactly one engine failure?
(b) Exactly two engine failures?
(c) More than two engine failures? (In this event, the airplane will crash.)
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§8.4 The Binomial Distribution
Example (c.f. Exercise 33 in the text). A biology quiz consists of eight multiple-choice
questions. Each question has five choices, of which only one is correct. If a student randomly
guesses on each question, what is the probability of getting more than 2 but less than 7
questions correct?
In previous sections, we considered the mean (or expected value), variance, and standard
deviation of an arbitrary random variable. It turns out that it is quite easy to compute these
statistics for binomial random variables.
If X is a binomial random variable associated to a binomial experiment consisting of n trials
with probability of success p, then
E(X) = np
V ar(X) = np(1 − p)
p
p
σX = V ar(X) = np(1 − p)
Example (Exercise 58 in the text). At a certain university, the probability that an entering
freshman will graduate within 4 years is .6. From an incoming class of 2000 freshmen, find:
(a) The expected number of students who will graduate within 4 years.
(b) The standard deviation of the number of students who will graduate within 4 years.