Lecture 11 – End of Chapter 17 Probability
Models & Chapter 18 Sampling Distributions
We did a simulation by flipping coins.
SORRY ABOUT THE RUSH AT THE END!
The Binomial Model
* Number of trials is n
n = 3 or n = 6 (depending on directions)
* Each trial is independent
on each flip the probabilities ‘re-set’
* Only 2 outcomes
Head or Tail
* Probability of a success on each trial is p (a
probability)
p = ½ = .5
* We are interested in X=number of successes
COUNT THE NUMBER OF HEADS
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Discovering the Characteristics of 1) a Binomial and of 2) a Geometric Model
Q. In the real world a coin flip could imitate what event?
Answer:
Experiment: Toss your coin 3 times and record each toss as either
Head (H) or Tail (T). Do your experiment 8 times.
#1___ ___ ___ #2 ___ ___ ___ #3 ___ ___ ___ #4 ___ ___ ___
1. How many heads?
2. On which toss did you
get the first head?
1. How many heads?
2. On which toss did you
Get the first head?
1. How many heads?
2. On which toss did you
Get the first head?
1. How many heads?
2. On which toss did you
Get the first head?
#5___ ___ ___ #6 ___ ___ ___ #7 ___ ___ ___ #8 ___ ___ ___
Describe the outcomes
Total number of Heads
In 3 tosses
Frequency (How many
experiments showed
that number of Heads?)
Prob
(Freq/
Total)
0
1
2
3
Total:
1.0
What classmates counts or probabilities seem
correct?
2
The Probability of exactly k successes in n trials is
nCk*pk*(1-p) n-k
nCk = n!/(k!*(n-k)!) where, for example, 4!=4*3*2*1
Theoretical
Describe the outcomes
Total number of Heads
In 3 tosses
Frequency (How many
experiments showed
that number of Heads?)
0
1
2
3
Total:
One Simulated In-class Example
Describe the outcomes
Total number of Heads
In 3 tosses
Frequency (How many
experiments showed
that number of Heads?)
Prob
(Freq/
Total)
.125
.375
.375
.125
1.000
Prob
(Freq/
Total)
0
1
2
3
Total:
Expected Value (from simulated):
Variance (from simulated):
3
1.0
Expected Value (from simulated):
Variance (from simulated):
mean of X (average number of successes in n
trials) is calculated as n*p =
And the variance is n*p*(1-p) =
How close?
WHAT DO YOU NEED TO KNOW?
1. Be able to identify a situation as Binomial
2. Pick out what n and p are
3. Calculate the chance of k(given) successes
in n (given trials) when you have p(given or
easily calculated, like coin flip)
4. Find the mean (n*p) and
variance n*p*(1-p)
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Example:
I am going to flip a coin 6 times (the other inclass example)
1. Is it Binomial?
2. What is n? What is p?
3. What is the prob of exactly 1 Head
In 6 tosses? (could ask 0 and 2 and 3…)
4. mean=? Variance=?
Which of the classmates simulations seem to be
correct?
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The Geometric Model
* Conduct trials : we flipped a coin
* Each trial is independent : yes
* Only 2 outcomes : Head or Tail
* Probability of a success on each trial is p (a
probability)
p= ½ = .5
* Interested in first trial when a success occurs
like, when did we get the first head?
WHAT DO YOU NEED TO KNOW?
1. Be able to identify a situation as Geometric
2. Pick out p is
3. Calculate the chance of the first success
happens on the k (given) trial.
The probability that the first success happens
on the 3rd trial (k=3) is
(1-p) k-1 *p
4. Find the mean (1/p) and
variance (1-p)/(p*p)
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Discovering the Characteristics of 2) a Geometric Model
Q. In the real world a wait for a head on coin flip could imitate what
event?
Answer:
Experiment: Toss your coin 3 times and record each toss as either
Head (H) or Tail (T). Do your experiment 8 times.
#1___ ___ ___ #2 ___ ___ ___ #3 ___ ___ ___ #4 ___ ___ ___
2. On which toss did you
get the first head?
2. On which toss did you
Get the first head?
2. On which toss did you
Get the first head?
2. On which toss did you
Get the first head?
#5___ ___ ___ #6 ___ ___ ___ #7 ___ ___ ___ #8 ___ ___ ___
Describe when
The first Head
appears
Frequency (How many
experiments showed
that number of Heads?)
Prob
(Freq/
Total)
1
2
3
4+
Total:
1.0
What classmates counts or probabilities seem
correct?
7
Problem: A claim is that 65% of AASU
students have ssn beginning with the number 2.
I might guess then that 65% of the students in
here do. I am going to “randomly” ask 5 of
you. What is the probability that exactly 2 of
the 5 have 2 as 1st ssn? How many do you
expect to have this (mean)? And what is the
variance associated with the number of 2ers?
ANS.
Could this be Binomial?
<if drawing from n items from N
make sure n/N < .10….that is,
without replacement will not create
massive dependence if you select less
than 10% to inspect=trial>
Any problem with independence ? n/N
Exactly 2 of 5?
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Mean?
Variance? Standard Deviation?
Q. If I selected a different 5 students would I
get the same answer? And then a different 5?
Use (#with 2)/5 as the sample’s proportion (call
it p-hat p̂ ; note that it is the probability of
finding such a person) of those whose ssn
begins with 2. This will vary depending on
which 5 students I randomly select.
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Note that p̂ is a mean=average since it is the
sum of raw data (0 or 1) divided by total number of data
points. Note also that we theoretically know everything
about the population (I said 65% had a 2 at start of ssn).
So, the CENTRAL LIMIT THEOREM applies
Check these criteria:
1. Taking a large sample (n)? “The larger the
better.” n = 30 usually works.
2. Make sure each sample=item=trial=person
independent from the other.
Then my mean looks like many other possible
means from different samples consisting of ‘n’
number of items and in fact comes from a
normal (bell-shaped) distribution (nearly) and
centers about the entire population mean with a
spread away (standard deviation=standard
error) equal to the population mean divided by
the square root of the sample size n.
x
Notation:
Is N( n 
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What about p̂ ?
Normal centered at p (population
percentage=proportion) with spread of
p *(1  p)
so long as n*p and n*(1-p)>10
n

Finally, let’s use the CLT on our ssn problem:
It is modeled by a Binomial (because we noted
that the 5 we picked was less than 10% of
the AASU population).
p = .65 (65%) n = 5; note 5*.65 and 5*.35=1.65
oops not greater than 10…anyway,
so our sample proportion p̂ = _________
is it near .65? spread away by
p *(1  p)
.65*.35

stdev)
n
5
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Think normal distribution
We call these sampling distributions = distributions
that would result from repeatedly calculating a mean
from this sample, then a different one, and another…
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