Transcript Slide 1

EXAM 1 IS…
1.
2.
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4.
Oct. 8th
Oct. 9th
Oct. 10th
Oct. 11th
25%
25%
25%
25%
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COMING UP

Homework 7, due Sunday

Exam 1, Oct. 10th
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CHAPTER 17
Copyright © 2008 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley
Probability Models
NORMAL PROBABILITY MODEL

Outcomes follow a distribution that is symmetric
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OTHER PROBABILITY MODELS
Geometric
 Binomial
 Poisson


All of these models are based on a Bernoulli Trial
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BERNOULLI TRIALS
The basis for the probability models we will
examine in this chapter is the Bernoulli trial.
 We have Bernoulli trials if:
 there are two possible outcomes (success and
failure).
 the probability of success, p, is constant.
 the trials are independent.

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INDEPENDENCE
One of the important requirements for Bernoulli
trials is that the trials be independent.
 When we don’t have an infinite population, the
trials are not independent. But, there is a rule
that allows us to pretend we have independent
trials:
 The 10% condition: Bernoulli trials must be
independent. If that assumption is violated, it
is still okay to proceed as long as the sample is
smaller than 10% of the population.


5 vs 5,000
YOU WISH TO DETERMINE THE OUTCOME OF A
VOTE FOR STUDENT PRESIDENT. ONLY JOHN
AND JANE ARE RUNNING. YOU POLL 10,000 OF
THE 20,000 STUDENTS. IS A BERNOULLI TRIAL
MODEL APPROPRIATE?
1.
2.
3.
4.
No, because there are more than 2 outcomes
Yes, because these trials may be considered Bernoulli
trials since the sample is less than 10% of the
population
Yes, because all 3 requirements for a Bernoulli trial
are met
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No, because the trials are not independent
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EXAMPLE

Flipping a Weighted Coin
40% Heads
 p=.4

Bernoulli Model
 X=probability of getting heads
 E(X) = p= 0.4
 Var(X)=p*(1-p) =0.24

PROBABILITY MODELS AND EXPECTED
VALUES

Based on the characteristics of these models
(independent trials and success/failure), we can
easily calculate the expected value and standard
deviation, because some other mathematician
has already done the work for us…




Bernoulli (success)
Geometric (how many until success)
Binomial (how many success in X trials)
Poisson (how many successes in X trails, special
case*)
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THE GEOMETRIC MODEL
A single Bernoulli trial accounts for only one
trial. If we have several attempts or repeated
trials, we need to account for the number of
attempt/trials.
 A Geometric probability model tells us the
probability for a random variable that counts the
number of Bernoulli trials until the first success.
 Geometric models are completely specified by one
parameter, p, the probability of success, and are
denoted Geom(p).

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THE GEOMETRIC MODEL (CONT.)
Geometric probability model for Bernoulli trials:
Geom(p)
p = probability of success
q = 1 – p = probability of failure
X = number of trials until the first success occurs
x-1
P(X = x) = q p
1
E(X)   
p

q
p2
THE GEOMETRIC MODEL


A Geometric probability model tells us the
probability for a random variable that counts the
number of Bernoulli trials until the first success.
What is the number of trials until the first
success?
X – number of trials until the first success
 E(X) = 1/p = 2.5
 Var(X)=(1-p)/p2 =3.75
 SD(X)=1.94

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BASKETBALL PROBLEM

A basketball player has made 80% of his foul
shots during the season. Assuming the shots are
independent, find the probability that in tonight’s
game he does the following…
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WHAT’S THE PROBABILITY THAT HE MISSES FOR
THE FIRST TIME ON HIS THIRD ATTEMPT?
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2.
3.
4.
(.8)*(.2)2
(.8)2*(.2)
(.8)*(.2)3
(.8)3*(.2)
0%
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0%Slide
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WHAT’S THE PROBABILITY THAT HE MAKES
HIS FIRST BASKET ON HIS FIFTH SHOT?
1.
2.
3.
4.
(0.8)*(0.2)5
(0.8)5*(0.2)
(0.8)*(0.2)4
(0.8)4*(0.2)
0%
0%
0%
0%Slide
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WHAT’S THE PROBABILITY THAT HE MAKE HIS
FIRST BASKET ON ONE OF HIS FIRST 3 SHOTS?
1.
2.
3.
4.
(0.8)*(0.2)2
(0.8)2*(0.2)
(0.8) 0 *(0.2) + (0.8) 1 *(0.2) + (0.8) 2*(0.2)
(0.8)*(0.2) 0+ (0.8)*(0.2) 1+ (0.8)*(0.2) 2
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EXPECTED VALUES AND THE GEOMETRIC
MODEL

Example, blood problem

In a group of 1,000 people, only 2% have Type AB
blood.
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ON AVERAGE, HOW MANY DONORS MUST BE
CHECKED TO FIND SOMEONE WITH TYPE AB
BLOOD?
1.
2.
3.
4.
1/0.02
1/0.98
0.02
0.98
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WHAT’S THE PROBABILITY THAT THERE IS A
TYPE AB DONOR AMONG THE FIRST 3 PEOPLE?
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2.
3.
4.
(0.02)*(0.98)3
(0.02)*(0.98)2
(0.02)*(0.98)0+(0.02) *(0.98)1+(0.02) *(0.98)2
(0.02) 0* (0.98) +(0.02) 1* (0.98) +(0.02) 2* (0.98)
0%
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0%Slide
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WHAT THE PROBABILITY THAT THE DONOR
WILL BE FOUND IN THE FIRST 6 PEOPLE?
1.
2.
3.
4.
Part B+(0.02)*(0.98)3+(0.02) *(0.98)4
+(0.02) *(0.98)5
Part B+(0.02) 3* (0.98) +(0.02) 4* (0.98)
+(0.02) 5* (0.98)
(0.02)*(0.98)5
(0.02) 5*(0.98)
0%
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WHAT IS THE PROBABILITY THAT THERE WON’T
BE A DONOR BEFORE THE 7TH PERSON?
1.
2.
3.
4.
(0.02) 6*(0.98)
(0.02) *(0.98) 6
1- (0.02) 6*(0.98)
1-part C
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THE GEOMETRIC MODEL

A Geometric probability model tells us the
probability for a random variable that counts the
number of Bernoulli trials until the first success.
p = probability of success
q = 1 – p = probability of failure
X = number of trials until the first success occurs
x-1
P(X = x) = q p
1
E(X)   
p

q
p2
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THE BINOMIAL MODEL
A Binomial model tells us the probability for a
random variable that counts the number of
successes in a fixed number of Bernoulli trials.
 Two parameters define the Binomial model: n,
the number of trials; and, p, the probability of
success. We denote this Binom(n, p).

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THE BINOMIAL MODEL (CONT.)

In n trials, there are
n!
n Ck 
k ! n  k !
ways to have k successes.


Read nCk as “n choose k.”
Note: n! = n x (n-1) x … x 2 x 1, and n! is read as “n
factorial.”
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THE BINOMIAL MODEL (CONT.)
Binomial probability model for Bernoulli trials:
Binom(n,p)
n = number of trials
p = probability of success
q = 1 – p = probability of failure
X = number of successes in n trials
n!
 n  x n x
n
P( X  x)    p q where  
 x
 x  x !(n  x)!
  np
  npq
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THE BINOMIAL MODEL

A Binomial model tells us the probability for a
random variable that counts the number of
successes in a fixed number of Bernoulli trials.
Weighted Coin p=.4 for a Heads
 n=3 trials
 X=number of success in the n trials
 E(X)=1.2
 Var(X)=0.72 and SD(X)=0.84

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DISTINGUISH BETWEEN GEOMETRIC AND
BINOMIAL

Example- Lefthanded people – problem
Assume that 10% of people are left-handed.
 6 people are selected at random.

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THE PROBABILITY THAT THE FIRST LEFTY IS
THE FIFTH PERSON – BINOMIAL OR
GEOMETRIC?
1.
2.
Binomial model
Geometric model
50%
50%
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2
WHAT’S THE PROBABILITY THAT THE FIRST
LEFTY IS THE FIFTH PERSON?
1.
2.
3.
4.
(.10)*(.90)5
(.10)5*(.90)
(.10)*(.90)4
(.10)4*(.90)
0%
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0%Slide
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THE PROBABILITY THERE ARE SOME LEFTIES
AMONG THE 6 PEOPLE – BINOMIAL OR
GEOMETRIC?
1.
2.
Binomial model
Geometric model
50%
50%
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1
2
WHAT’S THE PROBABILITY THERE ARE
SOME LEFTIES AMONG THE 6 PEOPLE?
1.
2.
3.
4.
1- (6 choose 0) * (0.1)6 *(0.9)0
1- (6 choose 0) * (0.1)1 *(0.9)5
1- (6 choose 0) * (0.1) 5 *(0.9)1
1- (6 choose 0) * (0.1)0 *(0.9) 6
0%
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0%Slide
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THE PROBABILITY THAT THE FIRST LEFTY IS
THE SECOND OR THIRD PERSON – BINOMIAL OR
GEOMETRIC?
1.
2.
Binomial model
Geometric model
50%
50%
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1
2
WHAT’S THE PROBABILITY THAT THE FIRST
LEFTY IS THE SECOND OR THIRD PERSON?
1.
2.
3.
4.
(0.1)2 * (0.9)+(0.1) 3 * (0.9)
(0.1)1 * (0.9)+(0.1) 2 * (0.9)
(0.1) * (0.9) 2 +(0.1) * (0.9) 3
(0.1) * (0.9) 1 +(0.1) * (0.9) 2
0%
0%
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0%Slide
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THE PROBABILITY THAT THERE ARE EXACTLY
LEFTIES IN THE GROUP – BINOMIAL OR
GEOMETRIC?
1.
2.
3
Binomial model
Geometric model
50%
50%
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1
2
WHAT’S THE PROBABILITY THAT THERE ARE
EXACTLY 3 LEFTIES IN THE GROUP?
1.
2.
3.
4.
(6 choose 3) * (0.1)3 * (0.9)3
(6 choose 3) * (0.1)2 * (0.9)4
(6 choose 3) * (0.1)4 * (0.9)2
We do not have enough information.
0%
1
0%
0%
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THE PROBABILITY THAT THERE ARE AT LEAST
LEFTIES IN THE GROUP - BINOMIAL OR
GEOMETRIC?
1.
2.
Binomial model
Geometric model
50%
4
50%
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1
2
WHAT’S THE PROBABILITY THAT THERE
ARE AT LEAST 4 LEFTIES IN THE GROUP?
1.
2.
3.
4.
1- (6 choose 4) * (0.1)4 * (0.9)2
1-(6 choose 4) * (0.1)2 * (0.9)4
(6 choose 4) * (0.1)4 * (0.9)2
+ (6 choose 5) * (0.1)5 * (0.9)1
+(6 choose 6) * (0.1)6 * (0.9)0
(6 choose 4) * (0.1)2 * (0.9)4
+ (6 choose 5) * (0.1)1 * (0.9)5
+(6 choose 6) * (0.1)0 * (0.9)6
0%
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0%
0%
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3
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0%
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THE PROBABILITY THAT THERE ARE NOT MORE
THAN 2 LEFTIES IN THE GROUP - BINOMIAL OR
GEOMETRIC?
1.
2.
Binomial model
Geometric model
50%
50%
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1
2
WHAT’S THE PROBABILITY THAT THERE ARE
NOT MORE THAN 2 LEFTIES IN THE GROUP?
1.
2.
3.
4.
1- (6 choose 2) * (0.1)4 * (0.9)2
1-(6 choose 2) * (0.1)2 * (0.9)4
(6 choose 2) * (0.1)4 * (0.9)2
+ (6 choose 1) * (0.1)5 * (0.9)1
+(6 choose 0) * (0.1)6 * (0.9)0
(6 choose 2) * (0.1)2 * (0.9)4
+ (6 choose 1) * (0.1)1 * (0.9)5
+(6 choose 0) * (0.1)0 * (0.9)6
0%
1
0%
0%
2
3
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0%
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BERNOULLI, GEOMETRIC, BINOMIAL

Bernoulli – single trial

Geometric – number of trials until success

Binomial – number of success in n trials

Poisson – number of success with large n and
very small p
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COMING UP

Homework 7, due Sunday

Exam 1, Oct. 10th
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