Download 8.1 Binomial Distributions

Section 8.1
Binomial Distributions
AP Statistics
The Binomial Setting
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Each observation falls into one of just
two categories, which for convenience
we call “success” or “failure”
There are a fixed number n of
observations
The n observations are all independent.
The probability of success, call it p, is the
same for each observation.
AP Statistics, Section 8.1.1
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The Binomial Setting: Example
1.
2.
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4.
Each observation falls into one of just two
categories, which for convenience we call
“success” or “failure”: Basketball player at the
free throw.
There are a fixed number n of observations:
The player is given 5 tries.
The n observations are all independent: When
the player makes (or misses) it does not change
the probability of making the next shot.
The probability of success, call it p, is the same
for each observation: The player has an 85%
chance of making the shot; p=.85
AP Statistics, Section 8.1.1
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Shorthand
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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.
AP Statistics, Section 8.1.1
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Example
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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(
)
AP Statistics, Section 8.1.1
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Example

Deal 10 cards from a shuffled deck and count
the number “X” of red cards.
A “success” is a red card.

How would you describe this using “B” notation?

AP Statistics, Section 8.1.1
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Binomial Coefficient
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Sometimes referred
to as “n choose k”
For example: “I have
10 students in a
class. I need to
choose 2 of them.”
In these examples,
order is not
important.
n
n!
 
 k  k ! n  k  !
10 
10!
 
 2  2!10  2 !
10  9  8  7  6  5  4  3  2 1

 2 18  7  6  5  4  3  2 1
 45
AP Statistics, Section 8.1.1
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Binomial Coefficients on the
Calculator
AP Statistics, Section 8.1.1
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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.

What is the probability that 3 children are type
O?
AP Statistics, Section 8.1.1
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Binomial Probabilities
n k
nk
P( X  k )     p  1  p 
k
 
AP Statistics, Section 8.1.1
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Binomial Distributions on the calculator
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Corinne makes 75% of her free
throws.
What is the probability of making
exactly 7 of 12 free throws.
n k
nk
   p  1  p 
k 
B(n,p) with k successes
binompdf(n,p,k)
AP Statistics, Section 8.1.1
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Binomial Distributions on the calculator
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Corinne makes 75% of her free
throws.
What is the probability of making at
most 2 of 12 free throws.
B(n,p) with k successes
binomcdf(n,p,k)
AP Statistics, Section 8.1.1
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Corinne makes 75% of her
free throws.
What is the probability of
making at least 9 of 12 free
throws.
B(n,p) with k successes
binomcdf(n,p,k)
AP Statistics, Section 8.1.1
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Mean and Standard Deviation of a
Binomial Distribution
  np
  np 1  p 
AP Statistics, Section 8.1.1
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B(10,.5), N (5, 10*.5*.5)
AP Statistics, Section 8.1.1
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B(100,.5), N (50, 100*.5*.5)
AP Statistics, Section 8.1.1
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B(1000,.5), N (500, 1000*.5*.5)
AP Statistics, Section 8.1.1
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Example:

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 timeconsuming.” Suppose that in fact 60% of all
adults would “agree”. What is the probability
that 1520 or more of the sample “agree”.
AP Statistics, Section 8.1.1
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Normal Approximation of Binomial Distribution
As the number of trials n gets larger, the
binomial distribution gets close to a normal
distribution.
 Question: What value of n is big enough?
The book does not say, so let’s see how
the close two calculations are…

AP Statistics, Section 8.1.1
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Normal Approximations for Binomial Distributions
As a “rule of thumb,” we may use the
Normal Approximation when…
 np ≥ 10 and n (1 – p) ≥ 10

AP Statistics, Section 8.1.1
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
Here are some useful applications of the
binomcdf and binomcdf commands:

To find P(x = k), use binompdf(n,p,k)
To find P(x ≤ k), use binomcdf(n,p,k)
To find P(x < k), use binomcdf(n,p,k-1)
To find P(x > k), use 1-binomcdf(n,p,k)
To find P(x ≥ k), use 1-binomcdf(n,p,k-1)
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AP Statistics, Section 8.1.1
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Homework
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Binomial Worksheet
AP Statistics, Section 8.1.1
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