Download The Binomial Probability Distribution

The Binomial Probability
Distribution
Section 6.2
Objectives
• Determine whether a probability experiment
is a binomial experiment
• Compute probabilities of binomial
experiments
• Compute the mean and standard deviation of
a binomial random variable
• Construct binomial probability histograms
Characteristics of Binomial
Probability Experiments
• Binomial Probability Distribution:
– Discrete
– Describes probabilities for experiments in which
there are only 2 mutually exclusive (disjoint)
outcomes
– Examples: yes/no, heads/tails,
Criteria for Binomial
Probability Experiments
• An experiment is said to be a binomial
experiment if:
– Performed a fixed number of times
• Each time is called a trial
– The trials are independent
• The outcome of one trial will not affect the probability of the
outcome of any of the other trials
– For each trial, there are only 2 mutually exclusive
(disjoint) outcomes: success or failure
– The probability of success is the same for each trial of
the experiment
Notation Used with the Binomial
Probability Distribution
• As always, a capital letter, X, is the notation
for a random variable
• There are always n independent trials of the
experiment
• Let p denote the probability of success, so that
1 – p is the probability of failure
• Let x denote the number of successes in n
independent trials of the experiment.
– So 0 < x < n
Examples
• Examples of binomial experiments
– Tossing a coin 30 times to see how many heads occur.
– Asking 100 people if they watch CNN news.
– Rolling a die to see if a 2 appears.
• Examples which aren't binomial experiments
– Rolling a die until a 4 appears (not a fixed number of
trials)
– Asking 50 people how old they are (not two
outcomes)
– Drawing 5 cards from a deck for a poker hand (done
without replacement, so not independent)
http://people.richland.edu/james/lecture/m170/ch06-bin.html
Compute Probabilities of
Binomial Experiments
Compute Probabilities of
Binomial Experiments
Binomial Probability
Distribution Function
Using the Binomial Probability
Distribution Function
• A coin is tossed 10 times. What is the probability that
exactly 6 heads will occur?
Using the Binomial Probability
Distribution Function
• A coin is tossed 10 times. What is the probability that
at least 5 heads will occur?
Using the Binomial Probability
Distribution Function
• A coin is tossed 10 times. What is the probability that
at fewer than 3 heads will occur?
Using the Binomial Probability
Distribution Function
• A coin is tossed 10 times. What is the probability that
3 - 5 heads will occur?
Assignment
• Pg 340: 1, 3, 4, 5, 6, 7-16, 17, 20, 23, 25, 28
Computing Probabilities Using the Binomial Table
A coin is tossed
10 times. What
is the probability
that exactly 6
heads will
occur?
http://images.tutorvista.com/cms/images/67/binomial-distribution-table.jpg
Computing Using the TI-83/84
http://cfcc.edu/faculty/cmoore/TIBinomial.htm
Computing Using the TI-83/84
http://cfcc.edu/faculty/cmoore/TIBinomial.htm
Computing Using the TI-83/84
http://cfcc.edu/faculty/cmoore/TIBinomial.htm
Compute the Mean and Standard
Deviation of a Binomial Random Variable
• These formulas are very simple and therefore
there are no commands on the TI-83/84
calculator to replicate this.
Example
According to the Experian Automotive, 35% of all car-owning
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
 (400)(0.35)
 140
 X  np(1  p)
 (400)(0.35)(1  0.35)
 9.54
Construct Binomial
Probability Histograms
x
0
1
2
3
4
5
6
7
8
9
10
P(x)
• Construct a binomial probability
histogram with n=10 and p=0.2. What
is the shape of this histogram?
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.
n = 8 and p = 0.15
n = 8 and p = 0. 5
n = 8 and p = 0.85
n = 50 and p = 0.8
n = 70 and p = 0.8
Number of Trials Affects Shape
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.
http://phet.colorado.edu/en/simulation/plinko-probability
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
 (400)(0.35)
 140
 X  2 X  140  2(9.54)
 120.9
 X  2 X  140  2(9.54)
 159.1
 X  np(1  p)
 (400)(0.35)(1  0.35)
 9.54
The result is unusual since 162 > 159.1
Assignment
• Pg 340-343: 29, 30, 34, 36, 42, 43, 45, 47, 54