Define a
binomial distribution
.
Explain when it might be all right to assume a binomial setting even though the independence condition is not
satisfied.
Explain what is meant by the
sampling distribution of a count
.
State the mathematical expression that gives the value of a
binomial coefficient
. Explain how to find the value of
that expression.
State the mathematical expression used to calculate the value of
binomial probability
.
Construction Objectives:
Students will be able to:
Evaluate a binomial probability by using the mathematical formula for
P
(
X = k
).
Explain the difference between
binompdf(n, p, X)
and
binomcdf(n, p, X)
.
Use your calculator to help evaluate a binomial probability.
If
X
is
B
(n,
p
), find µ
x
and
σ
x
(that is, calculate the mean and variance of a binomial distribution).
Use a
Normal approximation for a binomial distribution
to solve questions involving binomial probability.
Vocabulary:
Binomial Setting – random variable meets binomial conditions
Trial – each repetition of an experiment
Success – one assigned result of a binomial experiment
Failure – the other result of a binomial experiment
PDF – probability distribution function; assigns a probability to each value of X
CDF – cumulative (probability) distribution function; assigns the sum of probabilities less than or equal to X
Binomial Coefficient – combination of k success in n trials
Factorial – n! is n
×
(n1)
(n2)
…
2
1
Key Concepts:
Criteria for a Binomial Setting
A random variable is said to be a binomial provided:
1. The experiment is performed a
fixed number of times
.
Each
repetition is called a trial.
2. The trials are
independent
3. For each trial there are
two mutually exclusive
(disjoint)
outcomes
:
success or failure
4. The
probability of success is the same
for each trial of the
experiment
Most important skill for using binomial distributions is the
ability to recognize situations to which they do and don’t apply
Binomial PDF
The probability of obtaining
x
successes in
n
independent trials
of a binomial experiment, where the probability of success is
p
,
is given by:
P(x) =
n
C
x
p
x
(1 – p)
nx
,
x = 0, 1, 2, 3, …, n
n
C
x
is also called a
binomial coefficient
and is defined by
combination of
n
items taken
x
at a time
or
where n! is n
(n1)
(n2)
…
2
1
n
n!
=

k
k! (n – k)!
Mean and Standard Deviation:
n
TI83 Support:
•
For P(X = k) using the calculator:
2
nd
VARS binompdf(n,p,k)
•
For P(k ≤ X) using the calculator: 2
nd
VARS binomcdf(n,p,k)
•
For P(X ≥ k) use 1 – P(k < X) = 1 – P(k1 ≤ X)
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View Full DocumentChapter 8:
The Binomial and Geometric Distributions
Example 1:
Does this setting fit a binomial distribution? Explain
a)
NFL kicker has made 80% of his field goal attempts in the past.
This season he attempts 20 field goals.
The
attempts differ widely in distance, angle, wind and so on.
b)
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 Fall '12
 SonjaCox
 Geometric Probability, Binomial, Probability, AP Statistics, Probability theory, Cumulative distribution function, B. Geometric, Geometric distributions

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