stat 155, section 2, last time probability theory foundations of probability –events, sample space...

Stat 155, Section 2, Last Time

• Probability Theory

• Foundations of Probability– Events, Sample Space

– Probability Function

• Simple Random Sampling (count samples)

• Big Rules of Probability:– Not Rule ( 1 – P{opposite})– Or Rule

Reading In Textbook

Approximate Reading for Today’s Material:

Pages 259-271 , 311-323

Approximate Reading for Next Class:

Pages 277-286, 291-305

Midterm I

Coming up: Tuesday, Feb. 27

Material: HW Assignments 1 – 6

Extra Office Hours:

Mon. Feb. 26, 8:30 – 12:00, 2:00 – 3:30

(Instead of Review Session)

Bring Along:

1 8.5” x 11” sheet of paper with formulas

Midterm I

How will I test for Excel skills?

• No computers allowed

• Fill out menus with pencil

• Write Excel Commands with pencil

Put Excel commands (& details) on your:

1 8.5” x 11” sheet of paper with formulas

Midterm IExample: What fraction of N(1,2) population

is smaller than 0?

Could ask you to fill out menu:

You write:

0

1

2

true


is smaller than 0?

Above results in:

Note:

“command

line”


is smaller than 0?

So could ask you to simply write:

=NORMDIST(0,1,2,TRUE)

Note that you need to know:

• Excel function names

• Which arguments go where

So put all these on your sheet of formulas

(suggestion: make early & use to study)

Big Rules of Probability

1. Not Rule: P{not A} = 1 – P{A}

2. Or Rule:

P{A or B} = P{A} + P{B} – P{A and B}

Third rule?

Symbolic logic is based on:

and, or, not

How about a rule for and?


• Now head towards a rule for “and”

• Needs a new concept:

Conditional Probability

Idea: If event A is known to have occurred,

what is chance of B?

Note: “knowing A” means sample space is restricted to A


E.g. Roll a die, A = {even}, B = {1,2,3}

P{B, when A is known} = ???

(i.e. Somebody rolls, and only tells you

“even”. Note “<= 3” is no longer 50-50,

Since fewer even #s are <= 3)




Try “equally likely”:

CAREFUL: This is wrong!!!

Problem: for B, should not include 1 or 3,

since they are not even

133

##

AinBin




Correct Answer:

Makes sense, since chance should go down

from ½.

31

#,#

AinAinarethatBin


General definition:

Probability of B given A =

Next, by multiplying by P{A}, get and rule of

probability

}{

}&{

#

&#}|{

AP

ABP

Ain

ABinABP

And Rule of Probability

Big Rule III:

P{A & B} = P{A|B} P{B} = P{B|A} P{A}

Memory trick: like “canceling fractions”, but

make bar vertical, not a fraction

Note: 2 ways to do this. Good strategy: look

at both, as one is often easier.

The And Rule of Probability

HW:

4.89, 4.91a

4.95 (see 4.92)

And now for something completely different

Recall

Distribution

of majors of

students in

this course:

Stat 155, Section 2, Majors

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Busine

ss /

Man

.

Biolog

y

Public

Poli

cy /

Health

Pharm

/ Nur

sing

Jour

nalis

m /

Comm

.

Env. S

ci.

Other

Undec

ided

Fre

qu

ency


Three nurses died & went to heaven where they were met at the Pearly Gates by St. Peter.


To the first, he asked, "What did you do on Earth and why should you go to heaven?" "I was a nurse in an inner city hospital," she replied. "I worked to bring healing and peace to the poor suffering city children." "Very noble," said St. Peter. "You may enter." And in through the gates she went.


To the next, he asked the same question, "So what did you do on Earth?" "I was a nurse at a missionary hospital in Africa," she replied. "For many years, I worked with a skeleton crew of doctors and nurses who tried to reach out to as many peoples and tribes with a hand of healing and with a message of God's love." "How touching," said St. Peter. "You too may enter." And in she went.


He then came to the last nurse, to whom he asked, "So, what did you do back on Earth?" After some hesitation, she explained, "I was just a nurse at an H.M.O." St. Peter pondered this for a moment, and then said, "Okay, you may enter also." "Whew!" said the nurse. "For a moment there, I thought you weren't going to let me in."


"Oh, you can come in," said St. Peter, "but you can only stay for three days..."


Example illustrating power (and use) of rules:

Toss a Coin:

if H take a ball from I: R R G G G

if T take a ball from II: R R G

Now study progressively harder problems…

Balls in Urns Example

H R R G G G T R R G

E.g. A:

P{R | H} = 2/5

(chance of R, if know got H)

Simple “equally likely” calculation (just

counting) works here

Related HW

HW: C13

A company makes 40% of its cars at factory A, and the rest at factory B. Factory A produces 10% lemons, and Factory B produces 5% lemons. A car is chosen at random. What is the probability that:

(a) It came from Factory A? (0.4)

(b) It is a lemon, if it came from Fact. A? (0.1)


H R R G G G T R R G

E.g. B: P{R & H} = ???

Try simple counting:

P{R & H} = ???

Caution: This is wrong!!!

Reason: balls are not equally likely.

41

8

2

#

#

total

HinR


H R R G G G T R R G

E.g. B: P{R & H} = ???

Correct Answer:

P{R & H} = P{H | R} P{R} (OK, but hard)

= P{R | H} P{H} = (2/5)(1/2) = 1/5

Note: < ¼ (from wrong answer above)

Related HW

HW: C13

(c) It is a lemon, from Factory A? (0.04)

(think carefully about contrast with (b))


H R R G G G T R R G

E.g. C: P{R} = ???

Try simple counting:

P{R} = ???

Caution: This is wrong!!!

Reason: again balls are not equally likely.

21

84

## totalR


H R R G G G T R R G

E.g. C: P{R} = ???

Note: now expect > ½, since R’s in II are

more likely (thus get more weight)

Need to take which urn into account, so write

event in terms of the urn ball came from


H R R G G G T R R G

E.g. C: Correct Answer:

P{R} = P{(R & H) or (R & T)} = (“expand”)

= P{R & H} + P{R & T} – 0 (or Rule)

= 1/5 + P{R | T} P{T} = (from B)

= 1/5 + (2/3)(1/2) = 8/15

Note: slightly > ½ (as expected)

Related HW

HW: C13

(d) It is a lemon? (0.07)


H R R G G G T R R G

E.g. D: P{H | R} = ???

• Saved for last, since this is hardest

• Although only “turn around” of e.g. A

• This is common: One Cond. Prob. much

easier than the reverse


H R R G G G T R R G

E.g. D: P{H | R} =

Makes sense: if see R, less likely from H

}{}|{}{}|{

}{}|{

}{

}&{

TPTRPHPHRP

HPHRP

RP

RHP

83

533

21

32

21

52

21

52

Related HW

HW: C13

(e) It came from Factory A, if it is a lemon? (4/7)

4.103

4.105

Plotting Bivariate Data

Recall

Toy Example:

(1,2)

(3,1)

(-1,0)

(2,-1)

Toy Scatterplot, Separate Points

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 -1 0 1 2 3 4

x

y


Viewing Higher Dimensional Data:

• Extend to higher dimensions

• E.g. replace pairs by triples

• Make “3-d scatterplot”

• As “points in space”

• Think about “point cloud”


Toy

3-d

data

set:


High

Light

One

Point


X

Coor

of

High

Light


Y

Coor

of

High

Light


Z

Coor

of

High

Light


Proj-

ection

on

X

Axis


1-d

View:

Proj-

ection

on

X

Axis


Proj-

ection

on

Y

Axis


1-d

View:

Proj-

ection

on

Y

Axis


Proj-

ection

on

Z

Axis


1-d

View:

Proj-

ection

on

Z

Axis


Proj-

ection

on

X-Y

Plane


Proj-

ection

on

X-Y

Plane

rotated

up


Proj-

ection

on

X-Z

Plane


Proj-

ection

on

X-Z

Plane

rotated

up


Proj-

ection

on

Y-Z

Plane


Proj-

ection

on

Y-Z

Plane

rotated

up


Now

Look

At

All

Three


Put

Into

Single

Plot -

1d on

Diagn’l


Put

Into

Single

Plot -

2d off

Diagn’l


Called

Drafts-

man’s

Plot:

(study

3d

Objects

In 2d)

Recall Above Example

H R R G G G T R R G

E.g. D: P{H | R} =

Note: have “turned around” Cond. Probs…

}{}|{}{}|{

}{}|{

}{

}&{

TPTRPHPHRP

HPHRP

RP

RHP

83

533

21

32

21

52

21

52

Bayes Rule

Idea: Formal framework for turning around conditional probabilities

IF events are mutually exclusive and include everything

Set theoretically:

– intersections are empty

– union is sample space

– Called a “partition of the sample space”

kBB ,1

Bayes RuleIF events are mutually exclusive

and include everythingTHEN:

(decomposition of P{A} in terms of B’s)

Usefulness: turns around Cond. Probs.

So can write hard one in terms of easy ones

kBB ,1

}{}|{}{}|{}{}|{

}|{11

111

kk BPBAPBPBAPBPBAP

ABP

Bayes Rule

E.g. Balls & Urns, part D, above:

= Urn I (H)

= Urn II (T)

A = R (red ball)

Note: disjoint & includes everything

1B

2B

Bayes Rule Example

Disease Testing:

• Fundamental to modern medicine

• But most are not 100% accurate

• Study “Error Rate”

• Actually Error Rates, since 2 types of error

• Will see some surprises

(about turning around cond. probs.)

Disease Testing Example

Suppose 1% of population has a disease.

(fairly rare, but there are rarer diseases)

Tests are calibrated by applying to known cases:

Give test to 100 w/ Disease and 1000 Healthy

Suppose 80 have + reactions & 50 are +

What is “error rate”? (how good is the test???)

Disease Testing ExampleWhat is “error rate”?

Note: 2 types of “error”:

P{+ | H} = 50/1000 = 0.05

(Chance of healthy person called “sick”)

P{- | D} = (100 – 80) / 100 = 0.20

(Chance of sick person called “healthy”)

So “error rate” is ~ 20% or 5%?

(or something in between???)

Disease Testing ExampleCareful: We care about the opposite

conditional probabilities (turned around)

P{D | +}

I.e. IF have a + reaction

THEN what are chances of disease?

• Make much difference?

• Guess 80% or 95% (or in between)???

• Sell belongings and move to Bahamas???

Disease Testing ExampleApply Bayes Rule to turn around cond. probs.

Only ~14% !?! (what about 80% to 90%?)

HPHPDPDP

DPDPDP

|||

|

139.0)99.0)(05.0()01.0)(8.0(

)01.0)(8.0(

Disease Testing Example

Error rate only ~14% (unlikely have

disease?)

Reason 1: Rarity of disease magnifies errors

Reason 2: Test Population different from real

population

• View Bayes Rule Calculation as

adjustment for this

Bayes Rule HWC14: The workforce in a town has:

(20%, 50%, 30%)workers with

(no HS, HS-no C, C)education. Past experience indicates that

(10%, 30%, 90%)of workers with

(no HS, HS-no C, C)Education can perform a given task. Find the

probability that a randomly chosen worker:a. Can perform the task (0.44)b. Is College educated if (s)he can perform the task

(0.61)

stat 155, section 2, last time probability theory foundations of probability –events, sample space...

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