Basic Models of Probability
This presentation is the property of its rightful owner.
Sponsored Links
1 / 24

Basic Models of Probability PowerPoint PPT Presentation


  • 48 Views
  • Uploaded on
  • Presentation posted in: General

Basic Models of Probability. Instructor: Ron S. Kenett Email: [email protected] Course Website: www.kpa.co.il/biostat Course textbook: MODERN INDUSTRIAL STATISTICS, Kenett and Zacks, Duxbury Press, 1998. Course Syllabus. Understanding Variability Variability in Several Dimensions

Download Presentation

Basic Models of Probability

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Basic models of probability

Basic Models of Probability

Instructor: Ron S. Kenett

Email: [email protected]

Course Website: www.kpa.co.il/biostat

Course textbook: MODERN INDUSTRIAL STATISTICS,

Kenett and Zacks, Duxbury Press, 1998

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Course Syllabus

  • Understanding Variability

  • Variability in Several Dimensions

  • Basic Models of Probability

  • Sampling for Estimation of Population Quantities

  • Parametric Statistical Inference

  • Computer Intensive Techniques

  • Multiple Linear Regression

  • Statistical Process Control

  • Design of Experiments

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

The Paradox of the Chevalier de Mere - 1

Game A

Success = at least one “1”

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

The Paradox of the Chevalier de Mere - 2

Game B

Success = at least one “1,1”

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

The Paradox of the Chevalier de Mere - 3

Game A

Game B

P (Success) = P(at least one “1”)

P (Success) = P(at least one “1,1”)

Experience proved otherwise !

Game A was a better game to play

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

The Paradox of the Chevalier de Mere - 4

The calculations of Pascal and Fermat

Game A

Game B

P (Failure) = P(no “1”)

P (Failure) = P(no “1,1”)

P (Success) = .518

P (Success) = .491

What went wrong before?

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

P(outcomes add up to “10”) =?

To add or to multiply ?

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

1,11,21,31,41,51,6

2,12,22,32,42,52,6

3,13,23,23,43,53,6

4,14,24,34,44,54,6

5,15,25,35,45,55,6

6,16,26,36,46,56,6

P(outcomes add up to “10”) 36 / 3 =

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Mutually Exclusive Events

Two events are mutually exclusive (or disjoint) if it is impossible

for them to occur together.

If two events are mutually exclusive, they cannot be independent

and vice versa.

Example:

A subject in a study cannot be both male and female, nor can

they be aged 20 and 30. A subject could however be both male

and 20, both female and 30.

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Independent Events

Two events are independent if the occurrence of one of the

events gives us no information about whether or not the other

event will occur; that is, the events have no influence on each other.

If two events are independent then they cannot be mutually

exclusive (disjoint) and vice versa.

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Example

Suppose that a man and a woman each have a pack of 52 playing

cards. Each draws a card from his/her pack. Find the probability

that they each draw the ace of clubs.

We define the events

A = 'the man draws the ace of clubs'

B = 'the woman draws the ace of clubs'

Clearly events A and B are independent so,

That is, there is a very small chance that the man and the woman

will both draw the ace of clubs.

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Conditional Probability

In many situations, once more information becomes available, we

are able to revise our estimates for the probability of further

outcomes or events happening. For example, suppose you go out

for lunch at the same place and time every Friday and you are

served lunch within 15 minutes with probability 0.9. However,

given that you notice that the restaurant is exceptionally busy, the

probability of being served lunch within 15 minutes may reduce to

0.7. This is the conditional probability of being served lunch

within 15 minutes given that the restaurant is exceptionally busy.

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

The usual notation for "event A occurs given that event B has

occurred" is A|B (A given B). The symbol | is a vertical line and

does not imply division. P(A|B) denotes the probability that event

A will occur given that event B has occurred already.

A rule that can be used to determine a conditional probability

from unconditional probabilities is

P(A|B) = P(A andB) / P(B)

where,

P(A|B) = the (conditional) probability that event A will occur given

that event B has occurred already

P(A andB) = the (unconditional) probability that event A and event

B occur

P(B) = the (unconditional) probability that event B occurs

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Binomial Distribution

X = Number of successes in n trials

n = 6, x = 0

n = 6, x = 1

n = 6, x = 3

0

x

n

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Binomial Distribution

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Binomial Distribution

1316201112

1012201615

101291812

111191114

13841312

1411141512

18137119

151181116

912121815

139151212

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Poisson Distribution

X = Number of occurrences of an event

events

x = 0

x = 2

x = 8

x

0

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Poisson Distribution

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Negative Binomial

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Normal Distribution

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Normal Distribution

N(0,1)

X P(<X)P(Xi< <Xi+1)

-3.00.0013500.004432

-2.90.0018660.005953

-2.80.0025550.007915

-2.70.0034670.010421

-2.60.0046610.013583

-2.50.0062100.017528

-2.40.0081980.022395

-2.30.0107240.028327

-2.20.0139030.035475

-2.10.0178640.043984

-2.00.0227500.053991

-1.90.0287170.065616

-1.80.0359300.078950

-1.70.0445650.094049

-1.60.0547990.110921

-1.50.0668070.129518

-1.40.0807570.149727

-1.30.0968000.171369

-1.20.1150700.194186

-1.10.1356660.217852

-1.00.1586550.241971

-0.90.1840600.266085

-0.80.2118550.289692

-0.70.2419640.312254

-0.60.2742530.333225

-0.50.3085380.352065

-0.40.3445780.368270

-0.30.3820890.381388

-0.20.4207400.391043

-0.10.4601720.396953

0.00.5000000.398942

X P(<X)P(Xi< <Xi+1)

0.00.5000000.398942

0.10.5398280.396953

0.20.5792600.391043

0.30.6179110.381388

0.40.6554220.368270

0.50.6914620.352065

0.60.7257470.333225

0.70.7580360.312254

0.80.7881450.289692

0.90.8159400.266085

1.00.8413450.241971

1.10.8643340.217852

1.20.8849300.194186

1.30.9032000.171369

1.40.9192430.149727

1.50.9331930.129518

1.60.9452010.110921

1.70.9554350.094049

1.80.9640700.078950

1.90.9712830.065616

2.00.9772500.053991

2.10.9821360.043984

2.20.9860970.035475

2.30.9892760.028327

2.40.9918020.022395

2.50.9937900.017528

2.60.9953390.013583

2.70.9965330.010421

2.80.9974450.007915

2.90.9981340.005953

3.00.9986500.004432

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

Normal Distribution

7.900611.51519.95429.44938.2387

10.47079.40419.351710.566410.9079

10.007712.51889.693710.075710.1616

10.28819.856010.00149.846711.5006

10.29829.60239.723811.54138.4595

9.237211.040812.89969.55909.1041

8.91709.77347.98448.348411.3703

10.626010.095211.40198.98429.3783

9.75747.93128.15669.93059.1158

8.643610.46899.335610.87887.8790

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

The t Distribution

(c) 2000, Ron S. Kenett, Ph.D.


Basic models of probability

The F Distribution

(c) 2000, Ron S. Kenett, Ph.D.


  • Login