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Chapter 8:The Binomial and
Geometric Distributions
Copyright © 2008 by W. H. Freeman & Company
The Practice of StatisticsThird Edition
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Random Phenomena with Two
Possible Outcomes
• Coin toss
– Heads/Tails
• Shooting a free throw.
– Make/Miss
• The birth of a child.
– Boy/Girl
• These are called Binomial Distributions
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A discrete random variable situation can be
a binomial setting, a geometric setting (to
be covered later in the chapter), or neither.
You need to be able to recognize which
situation it is.
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If data are produced in a binomial setting, then the
random variable X = number of success. This is called a
BINOMIAL RANDOM VARIABLE, and the
probability distribution of X is called a BINOMIAL
DISTRIBUTION.
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Caution• Binomial distributions are an important
class of discrete probability distributions.
• The most important skill for
using binomial distributions is
the ability to recognize
situations to which they do and
don’t apply!!!
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Examples
• Blood types
– If both parents carry genes for O and A type
blood, each child has probability .25 of
inheriting type O blood.
– If couple has 5 children, the count X is the
number of success of the 5 children that have
type O blood.
– n = 5 and p = .25
– B(5, .25)
http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html
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Example
• Dealing Cards
– Deal 10 cards and count the number of red
cards.
– Not independent
– If first card is red, it is more likely the second
card is black.
– Not a binomial distribution
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Binomial Distributions
in Statistical Sampling
• Important because we use this to make
inferences the proportion “p” of success in a
population.
• Choose an SRS of n = 10 switches from 10,000
switches. 10% of switches are bad so p = 0.1.
– Not a binomial setting.
– However, with large samples, not having
independence is not that big of a deal.
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Assume that airplane engines have probability 0.999 of performing
properly for an hour of flight.
350 engines that fly for an hour without failure has a
B(350, 0.999) distribution.
This assumes engines fail independently.
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Binomial Formulas
• We can find a formula for the probability
that a binomial random variable takes any
value by adding probabilities for the
different ways of getting exactly that many
successes in n observations.
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Inherit Blood Type
• Child has .25 probability of having type O
blood.
• Parents have 5 children
• What is probability of exactly 2 of them
having type O blood?
• n = 5 and p = .25
• We want P(X = 2)
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Binomial Formulas
• S = Success (what does this mean in this context?)
• F = Failure
• Find the probability of a specific 2 out of 5.
• As an example:
– SFSFF (1st and 3rd have type O blood)
• (.25)(.75)(.25)(.75)(.75) = (.25)2(.75)3
• How many possible combinations of 2 success and 3
failures are there?
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Blood Types
• There are 10 possible combinations of 2
Successes and 3 Failures.
– How do I know this?
– We will find out on the next slide!
• They all have the same probability.
• So: P(X=2) = 10(.25)2(.75)3 = .2637
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n = 5, k = 2
n! = 5*4*3*2*1 = 120
k! = 2*1 = 2
(n – k)! = (5 – 2)! = 3! = 3*2*1 = 6
120/(2*6) = 120/12 = 10 Combinations
You should learn
this – not on
formula sheet!
Read aloud "n choose k."
A binomial coefficient equals
the number of combinations
of k items that can be
selected from a set of n items.
OR:
On your calculator, enter n and then Math:PRB:nCr and then
enter k and then ENTER.
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Binomial Probability
Defective Switches
n = 10, and p = .1
Probability that no more than 1 switch fails.
P(X ≤ 1) = P(X = 1) + P(X = 0)
In this case n = 10 and k = 1
Use what you just learned to try to do this.
P(X = 1) = .3874 and P(X = 0) = .3487
P(X ≤ 1) = .3874 + .3487 = .7361
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Assignment
• Play with Binomial Distributions Applet at
http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html and see when it
becomes approximately normal.
• “How to Calculate Binomial Probabilities” Packet
• Calculator videos that you need to know:
Binomial coefficient (Combinations):
https://youtu.be/MgSitJ7Aqxg?list=PLkIselvEzpM7N8zVRRUl7V8aTdoTsJ919
Binomial formula (just watch until 1:19 for now): https://youtu.be/F6JBimUE43U?list=PLkIselvEzpM7N8zVRRUl7V8aTdoTsJ919
• Need a review?
Watch https://youtu.be/mKtNpY1ZEdw?list=PLC8478000586FA6F9
• Read Pages 517 – 526