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Basic Models of Probability

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Basic Models of Probability

Instructor: Ron S. Kenett

Email: ron@kpa.co.il

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.

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.

The Paradox of the Chevalier de Mere - 1

Game A

Success = at least one “1”

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

The Paradox of the Chevalier de Mere - 2

Game B

Success = at least one “1,1”

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

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.

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.

P(outcomes add up to “10”) =?

To add or to multiply ?

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

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.

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.

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.

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.

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.

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.

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.

Binomial Distribution

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

Binomial Distribution

1316201112

1012201615

101291812

111191114

13841312

1411141512

18137119

151181116

912121815

139151212

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

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.

Poisson Distribution

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

Negative Binomial

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

Normal Distribution

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

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.

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.

The t Distribution

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

The F Distribution

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