Transcript Discrete Prob. Distrib.

Section 6.2 The Binomial Probability Distribution
6-1
Recall:
 A Discrete random variable
can take whole (countable) values
 A Continuous random variable
can take on any value between two specified
values
Examples:
• Flipping a coin a number of times: the Number of heads observed is a discrete
variable.
• A fire department mandates that all fire fighters must weigh between 150 and 250
pounds. The weight of a fire fighter would be an example of a continuous variable; since
a fire fighter's weight could take on any value between 150 and 250 pounds.
Consequently: Two types of
Probability Distributions
 Discrete Probability Distribution
 Continuous Probability Distribution
A Discrete Probability Distribution:
The Binomial Distribution
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 the outcome of one
trial will not affect the outcome of the other trials.
3. For each trial, there are two mutually exclusive (or disjoint)
outcomes, success or failure.
4. The probability of success is fixed for each trial of the
experiment.
6-5
Notation Used in the
Binomial Probability Distribution
• There are n independent trials of the experiment.
• p denotes the probability of success so 1 – p is the
probability of failure.
• X is the binomial random variable that denotes the
number of successes in n independent trials of the
experiment. So, 0 < X < n.
EXAMPLE
Identifying Binomial Experiments
Which of the following are binomial experiments?
(a) A player rolls a pair of fair die 10 times. The number X of
7’s rolled is recorded.
Binomial experiment
(b) The 11 largest airlines had an on-time percentage of
84.7% in November, 2001 according to the Air Travel
Consumer Report. In order to assess reasons for delays, an
official with the FAA randomly selects flights until she finds
10 that were not on time. The number of flights X that need
to be selected
is recorded.
Not a binomial
experiment – not a fixed number of trials.
(c) In a class of 30 students, 55% are female. The
instructor randomly selects 4 students. The number X of
Not a binomial
experiment – the trials are not independent.
females selected
is recorded.
6-7
Computing a Binomial Probability
Value
6-8
EXAMPLE
Using the Binomial Probability Distribution Function
According to the Experian Automotive, 35% of all car-owning households
have three or more cars.
(a) In a random sample of 20 car-owning households, what is the
probability that exactly 5 have three or more cars?
P (5) 
20
C5 (0.35)5 (1  0.35) 205
 0.1272
(b) In a random sample of 20 car-owning households, what is the
probability that less than 4 have three or more cars?
P( X  4)  P( X  3)
 P(0)  P(1)  P(2)  P(3)
 0.0444
(c) In a random sample of 20 car-owning households, what is the probability
that at least 4 have three or more cars?
P( X  4)  1  P( X  3)
 1  0.0444
 0.9556
Computing the Mean and Standard
Deviation of a Binomial Probability
Distribution
6-10
EXAMPLE
Binomial
Finding the Mean and Standard Deviation of a
Random Variable
According to the Experian Automotive, 35% of all carowning households have three or more cars. In a simple
random sample of 400 car-owning households, determine
the mean and standard deviation number of car-owning
households that will have three or more cars.
 X  np
 X  np(1  p)
 (400)(0.35)
 (400)(0.35)(1  0.35)
 140
 9.54
6-11
Constructing Binomial Probability
Histograms
EXAMPLES
(a) Construct a binomial probability histogram with n = 8 and p = 0.15
(b) Construct a binomial probability histogram with n = 8 and p = 0. 5
(c) Construct a binomial probability histogram with n = 8 and p = 0.85
For each histogram, comment on the shape of the distribution.
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n=8, p=0.15
x
P(x)
0
0.272491
1
0.384693
2
0.237604
3
0.08386
4
0.018499
5
0.002612
6
0.00023
7
1.16E-05
8
2.56E-07
n=8, p=0.5
x
P(x)
0
0.003906
1
0.03125
2
0.109375
3
0.21875
4
0.273438
5
0.21875
6
0.109375
7
0.03125
8
0.003906
6-14
n=8, p=0.85
x
P(x)
0
2.56289E-07
1
1.16184E-05
2
0.000230432
3
0.002611567
4
0.018498596
5
0.083860304
6
0.237604195
7
0.384692506
8
0.272490525
6-15
For a Larger number of trials, n=25:
6-16
n=50:
n=70:
Demo
For a fixed probability of success, p, as the
number of trials n in a binomial experiment
increase, the probability distribution of the
random variable X becomes bell-shaped.
As a general rule of thumb, if np(1 – p) > 10,
then the probability distribution will be
approximately bell-shaped.
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EXAMPLE
Using the Mean, Standard Deviation and
Empirical Rule to Check for
Unusual Results
in a Binomial Experiment
According to the Experian Automotive, 35% of all car-owning households
have three or more cars. A researcher believes this percentage is higher
than the percentage reported by Experian Automotive. He conducts a
simple random sample of 400 car-owning households and found that 162
had three or more cars. Is this result unusual ?
 X  np
 X  np(1  p)
 (400)(0.35)
 (400)(0.35)(1  0.35)
 140
 9.54
 X  2 X  140  2(9.54)
 120.9
 X  2 X  140  2(9.54)
 159.1
The result is unusual since 162 > 159.1
(more than 2 standard dev. greater than the mean)