Section 8.1
Binomial Distributions
AP Statistics
The Binomial Setting
1.
2.
3.
4.
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
2
The Binomial Setting: Example
1.
2.
3.
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
3
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.
AP Statistics, Section 8.1.1
4
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)
AP Statistics, Section 8.1.1
5
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?

This is not a Binomial distribution because once
you pull one card out, the probabilities change.

AP Statistics, Section 8.1.1
6
Binomial Coefficient



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
7
Binomial Coefficients on the
Calculator
AP Statistics, Section 8.1.1
8
Binomial Probabilities
n k
nk
P( X  k )     p  1  p 
k 
AP Statistics, Section 8.1.1
9
Binomial Mean
  np
  np 1  p 
AP Statistics, Section 8.1.1
10
B(10,.5), N (5, 10*.5*.5)
AP Statistics, Section 8.1.1
11
B(100,.5), N (50, 100*.5*.5)
AP Statistics, Section 8.1.1
12
B(1000,.5), N (500, 1000*.5*.5)
AP Statistics, Section 8.1.1
13
Binomial Distributions
on the calculator






Binomial Probabilities
B(n,p) with k successes
binompdf(n,p,k)
Corinne makes 75% of
her free throws.
What is the probability of
making exactly 7 of 12
free throws.
binompdf(12,.75,7)=.1032
n k
nk
   p  1  p 
k 
12 
5
7
  .75  .25 
7 
AP Statistics, Section 8.1.1
14
Binomial Distributions
on the calculator






Binomial Probabilities
12 
12 
5
7
6
6
.75
.25

.75
.25


 


 
B(n,p) with k successes  7 
6 
binomcdf(n,p,k)
12 
12 
5
7
   .75 .25     .754 .258 
Corinne makes 75% of
5 
4 
her free throws.
12 
12 
3
9
   .75 .25     .752 .2510 
What is the probability of  3 
2 
making at most 7 of 12
12 
12 
1
11
   .75 .25     .750 .2512 
free throws.
1 
0 
binomcdf(12,.75,7)=.1576
AP Statistics, Section 8.1.1
15
Binomial Distributions
on the calculator






Binomial Probabilities
12 
12 


5
7
B(n,p) with k successes   .75  .25     .758 .254 
7 
8 
binomcdf(n,p,k)
Corinne makes 75% of  12  .759 .253  12  .7510 .252










her free throws.
9 
10 
What is the probability of 12 
12 
11
1
12
0

.75
.25

.75
.25




making at least 7 of 12
 
 
11 
12 


free throws.
1-binomcdf(12,.75,6)=
AP Statistics, Section 8.1.1
16
Binomial Simulations
Corinne makes 75% of her free throws.
 Simulate shooting 12 free throws.
 randBin(n,p) will do one simulation
 randBin(n,p,t) will do t simulations

AP Statistics, Section 8.1.1
17
Normal Approximation of
Binomial Distribution

Remember
  np
  np 1  p 
AP Statistics, Section 8.1.1
18
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
19
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 time-consuming.” 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
20
TI-83 calculator
nCDF(1520, 1E99, 1500, 24.495)
 P(X>1520)=.207

B(2500,.6) and P(X>1520)
 1-binomcdf(2500,.6,1519)
 .2131390887

AP Statistics, Section 8.1.1
21
Homework
Binomial Worksheet
 Study Guide 8.2

AP Statistics, Section 8.1.1
22