Download The Binomial Probability Distribution

The Binomial Probability
Distribution
§6.2
In this section, we will introduce a specific type of probability distribution called a
binomial probability distribution. This distribution is a discrete distribution that describes
probabilities for experiments in which there are two disjoint outcomes. These two
outcomes are generally referred to as success and failure.
OBJECTIVE 1: Determine Whether a Probability
Distribution Experiment is a Binomial Experiment
Criteria for a Binomial Probability Experiment
An experiment is said to be a binomial experiment if
1. The experiment is performed a fixed number of times. Each repetition of the experiment is
called a trial.
2. The trials are independent. This means that the outcome of one trial will not affect the outcome
of the other trials.
3. For each trial, there are two (and only two) mutually exclusive outcomes: success and failure.
4. The probability of success is the same for each trial of the experiment.
We will let the random variable X be the number of successes in n trials of a binomial
experiment. So, X is called a binomial random variable.
Notation Used in the Binomial Probability Distribution

There are n independent trials of the experiment.

Let p denote the probability of success for each trial and so then the probability for
failure for each trial is _________________.

Let X denote the number of successes in n independent trials of the experiment.
So,
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Example 1: Determine whether the following probability experiment represents a
binomial experiment.
(a) A poll of 1200 registered voters is conducted in which the respondents are asked to
answer “Yes” or “No” on whether they feel that Congress should reform Social
Security.
(b) For each trial, three cards are selected from a standard 52 card deck without
replacement. The number of cards whose suit is hearts is recorded. After each trial, the
cards are left out of the deck.
(c) A random sample of 30 cars in a used car lot is obtained and their mileages are
recorded.
(d) A baseball player who reaches base safely 30% of the time is allowed to bat until he
reaches base safely. The random variable is how many at-bats he needed to reach the
base safely.
SECTION 6.2
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OBJECTIVE 2: Compute Probabilities of Binomial
Experiments
Binomial Probability Distribution Function
The probability of obtaining x successes in n independent trials of a binomial
experiment is given by
( )
where
nCx
(
)
and p is the probability of success.
Example 2: A binomial probability experiment is conducted with the given parameters.
Compute the probability of x successes in n trials of the experiment.
(a)
(b)
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Example 3: According to the Information Please almanac, 80% of adult smokers started smoking
before turning 18 years old.
(a) In a random sample of 20 smokers 18 years or older, what is the probability that
exactly 8 of them started smoking before 18 years of age.
(b) In a random sample of 20 smokers 18 years or older, what is the probability that at
least 8 of them started smoking before 18 years of age.
(c) In a random sample of 20 smokers 18 years or older, what is the probability that
fewer than 8 of them started smoking before 18 years of age.
(d) In a random sample of 20 smokers 18 years or older, what is the probability that
between 7 and 9 of them, inclusive, started smoking before 18 years of age.
SECTION 6.2
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Practice: According to Mediamark Research Inc, 86% of all US households own a
cellular phone.
(a) In a random sample of 20 households, what is the probability that the number of
households owning a cell phone is between 15 and 17, inclusive?
(b) In a random sample of 20 households, what is the probability that fewer than 15
own a cell phone?
(c) In a random sample of 20 households, what is the probability that exactly 15 own a
cell phone?
(d) In a random sample of 20 households, what is the probability that at least 15 own a
cell phone?
SECTION 6.2
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OBJECTIVE 3: Compute the Mean and Standard Deviation
of a Binomial Random Variable
Mean (or Expected Value) and Standard Deviation of a Binomial
Random Variable
A binomial experiment with n independent trials and probability of success p has a
mean and standard deviation given by the formulas
and
√
(
)
The mean of a binomial random variable is interpreted as the expected number of
successes in n trials of the experiment.
Example 4: According to the Information Please almanac, 80% of adult smokers started
smoking before turning 18 years old.
(a) In a random sample of 10 adult smokers, determine the mean and standard
deviation number of adult smokers that started smoking before turning 18 years old.
(b) Interpret the mean.
Note: A binomial probability histogram is constructed the same way that any
probability histogram is constructed.
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OBJECTIVE 4: Using the Mean, Standard Deviation, and
Empirical Rule to Check for Unusual Results in a Binomial
Experiment
Binomial Distributions and the Empirical Rule

As the number of trials in a binomial experiment increases, the probability distribution
of the random variable X becomes bell shaped.
 As a general rule, if
(
)
, then the probability distribution will be
approximately bell shaped. This allows us to use the Empirical Rule to identify
unusual observations. (Recall that the cutoff point for unusual observations is if the
probability is 5%.)
to
Recall: In section 3.2, we were able to use the Empirical Rule for any distribution
that is roughly bell shaped.
SECTION 6.2
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Example 4 (continued): According to the Information Please almanac, 80% of adult
smokers started smoking before turning 18 years old. Suppose that we took a random
sample of 200 adult smokers, determine if it would be unusual if we found 180 smokers
that started smoking before 18 years of age.
Example 5: According to CTIA, 25% of all US households are wireless-only. In a simple
random of 300 households, 92 were wireless-only. Is this result unusual?
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Practice: According to Mediamark Research Inc, 86% of all US households in 2007
owned at least one cell phone. In a simple random sample of 300 households, 275
owned at least one cell phone. Is this result unusual?
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