Download The Binomial Distribution

8.1 The Binomial Distribution
1. What are the conditions for the Binomial
Setting?
2. What is the Binomial Distribution
Notation?
3. How do you compute the probability of a
value of X using the Binomial Distribution?
4. How do you find the means and standard
deviation of the Binomial Distribution?
The Binomial Distribution
Properties of a Binomial Experiment
1.
2.
3.
4.
It consists of a fixed number of observations
called trials.
Each trial can result in one of only two mutually
exclusive outcomes labeled success (S) and
failure (F).
Outcomes of different trials are independent.
The probability that a trial results in S is the same
for each trial.
The binomial random variable X is defined as
X = number of successes observed when
experiment is performed
The probability distribution of X is called the
binomial probability distribution.
n!
 
n Ck    
k
  k !(n  k )!
n
Shorthand
• Normal distributions can be described using the
N(µ,σ) notation; for example, N(65.5,2.5) is a
normal distribution with mean 65.5 and standard
deviation 2.5.
• Binomial distributions can be described using
the B(n,p) notation; for example, B(5, .85)
describes a binomial distribution with 5 trials and
.85 probability of success for each trial.
Example
• Blood type is inherited. If both parents carry
genes for the O and A blood types, each child
has probability 0.25 of getting two O genes and
so of having blood type O. Different children
inherit independently of each other. The number
of O blood types among 5 children of these
parents is the count X off successes in 5
independent observations.
• How would you describe this with “B” notation?
• X=B(5,.25)
The Binomial Distribution
Let
n = number of independent trials in a
binomial experiment
p = constant probability that any particular
trial results in a success.
Then
P( X  k )  P(k successes among n trials)
n!
nk
k

p 1  p 
k!n - k !
Example
The adult population of a large urban area is
60% black. If a jury of 12 is randomly selected
from the adults in this area, what is the
probability that precisely 7 jurors are black.
Clearly, n=12 and p =.6, so
p ( 7) 
12!
(.6) 7 (.4)5  0.2270
7! 5!
12! 1 2  3  4  5  6  7  8  9 10 1112
Note :

7! 5! (1 2  3  4  5  6  7)(1 2  3  4  5)
8  9 10 1112

 8  9 11  792
1 2  3  4  5
Example - continued
The adult population of a large urban area is
60% black. If a jury of 12 is randomly selected
from the adults in this area, what is the
probability that less than 3 are black.
Clearly, n = 12 and p = 0.6, so
P(x  3)  P(x  2)  p(0)  p(1)  p(2)
12!
12!
12!

(.6)0 (.4)12 
(.6)1(.4)11 
(.6)2 (.4)10
0!12!
1!11!
2!10!
 0.00002  0.00031  0.00249  0.00281
Another Example
On the average, 1 out of 19 people will
respond favorably to a certain telephone
solicitation. If 25 people are called,
a) What is the probability that exactly two
will respond favorably to this sales pitch?
n  25,
1
p
19
25! 1 2 18 23
p(2) 
( ) ( )  0.2396
2!23! 19 19
Mean & Standard Deviation of a
Binomial Random Variable
The mean value and the standard
deviation of a binomial random variable
are, respectively,
 X  np
 X  np(1  p)
Example
A professor routinely gives quizzes containing
50 multiple choice questions with 4 possible
answers, only one being correct.
Occasionally he just hands the students an
answer sheet without giving them the questions
and asks them to guess the correct answers.
Let x be a random variable defined by
x = number of correct answers on such an exam
Find the mean and standard deviation for x
Example - solution
The random variable is clearly binomial
with n = 50 and p = ¼. The mean and
standard deviation of x are
1
 X  np  50   12.5
4
 1  3 
 X  50    9.375  3.06
 4  4 
As n gets large
The next three slides will
show you a comparison of
the binomial distribution to
a normal density curve.
B(10,.5), N (5, 10*.5*.5)
B(100,.5), N (50, 100*.5*.5)
B(1000,.5), N (500, 1000*.5*.5)
Conclusion about the Normal
Distribution
• When n is large, the distribution of X is
approximately Normal.
• As a rule of thumb, we will use the Normal
approximation when n and p satisfy:
np ≥ 10 and n(1 – p) ≥ 10
Additional cautions with the Normal
approximation
• The accuracy of the Normal approximation
improves as the sample size n increases.
• It is most accurate when p is close to ½
and least accurate when p is near 0 or 1.
• It is not accurate when the distribution is
skewed.
• When to use the Normal approximation
depends on how accurate your
calculations need to be.