AP STATISTICS
LESSON 8 – 1
( DAY 2 )
THE BINOMIAL DISTRIBUTION
(BINOMIAL FORMULAS)
Binomial Formulas
Example 8.9
page 446
Each child born to a particular
set of parents has probability
0.25 of having blood type O.
If these parents have 5 children,
what is the probability that
exactly 2 of them have type O
blood?
ESSENTIAL QUESTION:
What are the binomial formulas
and how are they used to solve
problems that can be modeled in
binomial settings?
Objectives:
• To define and use the binomial
formulas.
• To derive and use binomial means
and standard deviation
Binomial Coefficient =
The number of ways of
arranging k successes among n
observations is given by the
binomial coefficient
n
k
=
n!
k!(n – k )!
for k = 0,1,2…..n.
Vocabulary
Factorial - ! – The formula for binomial
coefficients uses the factorial notation.
For any positive whole number n, its
factorial n! is
n! = n x (n-1) x ( n – 2 ) x …x 3 x 2 x1
The notation n is not related to the
k
n
Fraction k. A helpful way to remember its
meaning is to read it as “ binomial
coefficient “ n choose k.”
Binomial Probability
If X has the binomial
distribution with n observations
and probability p of success on
each observation, the possible
values of X are 0, 1, 2, 3, …n. If
k is any one of these values,
P(X = k ) = n pk(1 – p)n - k
k
Example 8.10
Defective Switches
Page 448
The number X of switches that
fail inspection in Example 8.3
has approximately the binomial
distribution with n = 10 and
p = 0.1.
Find the probability that no
more than 1 fails.
Mean and Standard
Deviation of a Binomial
Random Variable
If a count X has the binomial distribution
with number of observations n and
probability p, the mean and the standard
deviation of X are
μ = np
σ = √ np(1 – p )
These short formulas are good only for
binomial distributions. They can’t be used
for other discrete random variables.
The Normal Approximation
to Binomial Distributions
The formula for binomial probabilities
becomes awkward as the number of
trials n increases.
As the number of trials n gets larger,
the binomial distribution gets close to
normal distribution.
When n is large, we can use normal
probability calculations to approximate
binomial probabilities.
Example 8.12
Attitudes Toward Shopping
Page 452
Are attitudes toward shopping changing?
A recent survey asked a
nationwide random sample
of 2500 adults if they
agreed or disagreed that “I
like buying new clothes, but
shopping is often frustrating
and time-consuming.”
p = 60% and X ≤ 1520
Normal Approximation for
Binomial Distributions
Suppose that a count X has the binomial
distribution with n trials and success
probability p. When n is large, the
distribution of X ix approximately normal,
N(np,√np(1 – p)).
As a rule of thumb, we will use normal
approximation when n and p satisfy
np ≥ 10 and n(1-p) ≥ 10.