What is probability? Marco Cattaneo Department of Physics and - - PowerPoint PPT Presentation

SLIDE 1

What is probability?

Marco Cattaneo

Department of Physics and Mathematics, University of Hull Applicant Day 18 February 2015

SLIDE 2

SLIDE 3

SLIDE 4

SLIDE 5

SLIDE 6

SLIDE 7

SLIDE 8

statistical literacy

Marco Cattaneo @ University of Hull What is probability? 2/7

SLIDE 9

statistical literacy

◮ probability and statistics as high level numeracy skills

Marco Cattaneo @ University of Hull What is probability? 2/7

SLIDE 10

statistical literacy

◮ probability and statistics as high level numeracy skills ◮ expected of citizens in modern societies to understand news, product

information, political debate, etc.

Marco Cattaneo @ University of Hull What is probability? 2/7

SLIDE 11

statistical literacy

◮ probability and statistics as high level numeracy skills ◮ expected of citizens in modern societies to understand news, product

information, political debate, etc.

◮ increases employability:

“I keep saying the sexy job in the next ten years will be

• statisticians. People think I’m joking, but who would’ve guessed

that computer engineers would’ve been the sexy job of the 1990s? The ability to take data—to be able to understand it, to process it, to extract value from it, to visualize it, to communicate it—that’s going to be a hugely important skill in the next decades, not only at the professional level but even at the educational level for elementary school kids, for high school kids, for college kids. Because now we really do have essentially free and ubiquitous data. So the complimentary scarce factor is the ability to understand that data and extract value from it.”

Hal Varian (professor at UC Berkeley, chief economist at Google), McKinsey, January 2009

Marco Cattaneo @ University of Hull What is probability? 2/7

SLIDE 12

statistical literacy

◮ probability and statistics as high level numeracy skills ◮ expected of citizens in modern societies to understand news, product

information, political debate, etc.

◮ increases employability:

“I keep saying the sexy job in the next ten years will be

• statisticians. People think I’m joking, but who would’ve guessed

that computer engineers would’ve been the sexy job of the 1990s? The ability to take data—to be able to understand it, to process it, to extract value from it, to visualize it, to communicate it—that’s going to be a hugely important skill in the next decades, not only at the professional level but even at the educational level for elementary school kids, for high school kids, for college kids. Because now we really do have essentially free and ubiquitous data. So the complimentary scarce factor is the ability to understand that data and extract value from it.”

Hal Varian (professor at UC Berkeley, chief economist at Google), McKinsey, January 2009

Marco Cattaneo @ University of Hull What is probability? 2/7

SLIDE 13

mathematical definition

ERGEBNISSE DER MATHEMATIK UND IHRER GRENZGEBIETE

HERAUSGEGEBEN VON DER SCHRIFTLEITUNG DES "ZENTRALBLATT FQR MATHEMATIK" ZWEITER BAND

• -------------3--------------

GRUNDBEGRIFFE DER WAHRSCHEINLICHKEITScr RECHNUNG

VON

• A. KOLMOGOROFF

BERLIN VERLAG VON JULIUS SPRINGER 1933

Marco Cattaneo @ University of Hull What is probability? 3/7

SLIDE 14

mathematical definition

probability is a normalized measure

ERGEBNISSE DER MATHEMATIK UND IHRER GRENZGEBIETE

HERAUSGEGEBEN VON DER SCHRIFTLEITUNG DES "ZENTRALBLATT FQR MATHEMATIK" ZWEITER BAND

• -------------3--------------

GRUNDBEGRIFFE DER WAHRSCHEINLICHKEITScr RECHNUNG

VON

• A. KOLMOGOROFF

BERLIN VERLAG VON JULIUS SPRINGER 1933

Marco Cattaneo @ University of Hull What is probability? 3/7

SLIDE 15

mathematical definition

probability is a normalized measure Andrey Kolmogorov (Tambov 1903 – Moscow 1987): Grundbegriffe der Wahrscheinlichkeitsrechnung (1933)

ERGEBNISSE DER MATHEMATIK UND IHRER GRENZGEBIETE

HERAUSGEGEBEN VON DER SCHRIFTLEITUNG DES "ZENTRALBLATT FQR MATHEMATIK" ZWEITER BAND

• -------------3--------------

GRUNDBEGRIFFE DER WAHRSCHEINLICHKEITScr RECHNUNG

VON

• A. KOLMOGOROFF

BERLIN VERLAG VON JULIUS SPRINGER 1933

Marco Cattaneo @ University of Hull What is probability? 3/7

SLIDE 16

ERGEBNISSE DER MATHEMATIK

UND IHRER GRENZGEBIETE

HERAUSGEGEBEN VON DER SCHRIFTLEITUNG

DES

"ZENTRALBLATT FQR MATHEMATIK" ZWEITER BAND

• -------------3--------------

GRUNDBEGRIFFE DER WAHRSCHEINLICHKEITScr RECHNUNG

VON

• A. KOLMOGOROFF

BERLIN

VERLAG VON JULIUS SPRINGER 1933

SLIDE 17

FOUNDATIONS

OF THE

THEORY OF PROBABILITY

BY

A

• N o

KOLMOGOROV

Second Englirh Edition

TRANSLATION EDITED BY

NATHAN MORRISON

WITH

AN ADDED BlBLIOCRAPHY BY

• A. T. BHARUCHA-REID

UNIVERStTY OF OREGON

CHELSEA PUBLISHING COMPANY NEW YORK

SLIDE 18

2

I, Elementary Theory of Probability

the system of axioms and in the further development of the theory, then the postulational concepts of a random event and its probability seem the most suitable. There are other postula- tional systems of the theory of probability, particularly those in which the concept of probability is not treated as one of the basic concepts, but is itself expressed by means of other concept8.l However, in that caae, the aim is different, namely, to tie up as closely as possible the mathematical theory with the empirical development of the theory of probability. 8 1

. Axioms

Let S be a collection of elements

E, q, g, .

.

.

,

w h i c h

we shall call

elementary events, and 8 a set of subsets of E ;

the elements of

the set 8

will be called random events.

• I. 5

is a fielda

• f sets.

1 1 . id: contains the set E.

• 111. To

each set A in 8 is assigne& a

non-negative red number P ( A ) .

This number P ( A ) is called the probability of the event A.

IV.

P(E) equals 1.

V .

I f A and

B h v e m elemeat i

n common, them

A system of s

e h , 8, together with a definite assignment of

numbera P(A), satisfying Axioms I-V,

is called a field of prab-

ability.

Our

system of Axioms I-V

is consistent. This is proved by the

following example. Let E consist of the single element e and let 8 consist of E and the null set 0. P(E) is then set equal to 1 and

P(0) equals 0.

'

For example, R. von Misesf 1

]

and [2) and S. Bernatein [I].

a The

readex who wishes from the outset to give a concrete meaning to the following axioms, is referred to 8 2.

'

• Cf. HAUSDORFF,

Mengedehre, 1927, p. 78. A system of sets is called a field

if the sum, product, and difference of two seta of the system alao belong

to the same system. Every non-empty field contains the null set 0. Using Hausdorff's notation, we designate the roduct of A and B b AB;,the sum b A 4- B in

B

1 ;

&

the case where A B

=

0; an in the general

case y A +

B; the di erence of

A and 3

by A-B. The set E-A, which is the complement of A, will be denoted by A. We ahall assume that the reader is familiar with the fundamental rules

• f operations of Bets and their sums, products, and differences. All subsets
• f fJ will be designated by Latin capitals.

SLIDE 19

2 I. Elementary Theory of ProbabUlty

the system of axioms and in the further development of the theory, then the postulational concepts of a random event and its probability seem the most suitable. There are other postula- tional systems of the theory of probability, particularly those in which the concept of probability is not treated as one of the basic concepts, but is itself expressed by means of other concepts.' However, in that case, the aim is different, namely, to tie up as closely as possible the mathematical theory with the empirical development of the theory of probability. § 1. Axiom8 1

Let E be a collection of elements f, '1' C, ... , which we shall call eLemen,ta,ry events, and it a set of subsets of E; the elements of the set i'f will be called ra.ndom events.

• 1. if is a field" of sets.
• II. tl contains the set E.
• III. To each set A in iJ is a8signed a non-negative real number

peA). This number peA) is called the probability of the event A.

• IV. peE) equals 1.
• v. If A and B have no element in. common., then

P(A+B) =P(A) +P(B)

A system of sets, iJ, together with a definite assignment of

numbers peA), satisfying Axi-oms I-V, is called a field of prob- ability.

Our system of Axioms I-V is co'l'l,8istent. This is proved by the following example. Let E consist of the single element eand let ~

consist of E and the null set O. P(E) is then set equal to 1 and

P(O) equals O.

1 For example, R. von Mises[l] and [2] and S. Bernstein [1]. I The reader who wishes from the outset to give a concrete meaning to the

following axioms, is referred to § 2.

• Cf. HAUSDORFF, Mengemehre, 1927, p. 78. A system of sets is called a field

if the sum, product, and difference of two sets of the system also belong to the same system. Every non-empty field contains the null set o. Using Hausdorff's notation, we designate the product of A and B by AB; .the sum by A + B in the case where AB = 0; and in the general ease by A +B; the difference of A and B by A-B. The set E-A, which is the complement of A, will be denoted by A. We shall assume that the reader is familiar with the fundamental rules

• f operations of sets and their Bums, products, and differences. All subsets
• t S' will be designated by Latin capitals.

SLIDE 20

Chapter I ELEMENTARY THEORY OF PROBABILITY

We define as elementary theory of probability that part of the theory in which we have to deal with probabilities of only a finite number of events. The theorems which we derive here can be applied also to the problems connected with an infinite number

• f random events. However, when the latter are studied, emen-

tially new principles are used. Therefore the only axiom of the mathematical theory of probability which deals particularly with the case of an infinite number of random events is not introduced until the beginning of Chapter I1 (Axiom VI). The theory of probability, as a mathematical discipline, can and should be developed from axioms in exactly the same way

as Geometry and Algebra. This means that after we have defined

the elements to be studied and their basic relations, and have stated the axiome by which theae relations are to be governed, all further exposition must be based exclusively on these axioms, independent of the usual concrete meaning of these elements and their relations.

In accordance with the above, in 8 1 the concept of a field of

probabilities is defined aa a system of aets which satisfies certain

• conditions. What the elements
• f this aet represent is of no im-

portance in the purely mathematical development of the theory

• f probability (cf. the introduction of basic geometric concepts

in the Foundatiow of Geometry by Hilbert, or the definitions of groups, ring^ and fielda in abstract algebra). Every axiomatic (abstract) theory admits, as is well known,

• f an unlimited number of concrete interpretations besides those

from which it was derived. Thus we find applications in fields of science which have no relation to the concepts of random event and of probability in the precise meaning of theae words. The poatulational basis of the theory of probability can be established by different methods in respect to the selection of axioms as well as in the selection of basic concepts and relations. However, if our aim is to achieve the utmost simplicity both in

I

SLIDE 21

Chapter I ELEMENTARY THEORY OF PROBABILITY

We define as elementary theory of probability that part of the theory in which we have to deal with probabilities of only a finite number of events. The theorems which we derive here can be applied also to the problems connected with an infinite number

• f random events. However, when the latter are studied, emen-

tially new principles are used. Therefore the only axiom of the mathematical theory of probability which deals particularly with the case of an infinite number of random events is not introduced until the beginning of Chapter I1 (Axiom VI). The theory of probability, as a mathematical discipline, can and should be developed from axioms in exactly the same way

as Geometry and Algebra. This means that after we have defined

the elements to be studied and their basic relations, and have stated the axiome by which theae relations are to be governed, all further exposition must be based exclusively on these axioms, independent of the usual concrete meaning of these elements and their relations.

In accordance with the above, in 8 1 the concept of a field of

probabilities is defined aa a system of aets which satisfies certain

• conditions. What the elements
• f this aet represent is of no im-

portance in the purely mathematical development of the theory

• f probability (cf. the introduction of basic geometric concepts

in the Foundatiow of Geometry by Hilbert, or the definitions of groups, ring^ and fielda in abstract algebra). Every axiomatic (abstract) theory admits, as is well known,

• f an unlimited number of concrete interpretations besides those

from which it was derived. Thus we find applications in fields of science which have no relation to the concepts of random event and of probability in the precise meaning of theae words. The poatulational basis of the theory of probability can be established by different methods in respect to the selection of axioms as well as in the selection of basic concepts and relations. However, if our aim is to achieve the utmost simplicity both in

I

SLIDE 22

example: probability of rain

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 23

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 24

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow

◮ It will rain tomorrow in 30% of the Hull region.

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 25

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow

◮ It will rain tomorrow in 30% of the Hull region. ◮ It will rain tomorrow for 30% of the time.

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 26

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow

◮ It will rain tomorrow in 30% of the Hull region. ◮ It will rain tomorrow for 30% of the time. ◮ 30% of weather forecasters believe it will rain tomorrow.

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 27

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow

◮ It will rain tomorrow in 30% of the Hull region. ◮ It will rain tomorrow for 30% of the time. ◮ 30% of weather forecasters believe it will rain tomorrow. ◮ It will rain on 30% of the days like tomorrow.

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 28

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow

◮ It will rain tomorrow in 30% of the Hull region.

16%

◮ It will rain tomorrow for 30% of the time.

10%

◮ 30% of weather forecasters believe it will rain tomorrow.

22%

◮ It will rain on 30% of the days like tomorrow.

19%

Rebecca E. Morss, Julie L. Demuth, and Jeffrey K. Lazo: Communicating uncertainty in weather forecasts: A survey of the U.S. public, Weather and Forecasting 23 (2008)

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 29

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow (at least 0.1 mm at the weather station)

◮ It will rain tomorrow in 30% of the Hull region.

16%

◮ It will rain tomorrow for 30% of the time.

10%

◮ 30% of weather forecasters believe it will rain tomorrow.

22%

◮ It will rain on 30% of the days like tomorrow.

19%

Rebecca E. Morss, Julie L. Demuth, and Jeffrey K. Lazo: Communicating uncertainty in weather forecasts: A survey of the U.S. public, Weather and Forecasting 23 (2008)

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 30

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow (at least 0.1 mm at the weather station)

◮ It will rain tomorrow in 30% of the Hull region.

× 16%

◮ It will rain tomorrow for 30% of the time.

10%

◮ 30% of weather forecasters believe it will rain tomorrow.

22%

◮ It will rain on 30% of the days like tomorrow.

19%

Rebecca E. Morss, Julie L. Demuth, and Jeffrey K. Lazo: Communicating uncertainty in weather forecasts: A survey of the U.S. public, Weather and Forecasting 23 (2008)

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 31

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow (at least 0.1 mm at the weather station)

◮ It will rain tomorrow in 30% of the Hull region.

× 16%

◮ It will rain tomorrow for 30% of the time.

× 10%

◮ 30% of weather forecasters believe it will rain tomorrow.

22%

◮ It will rain on 30% of the days like tomorrow.

19%

Rebecca E. Morss, Julie L. Demuth, and Jeffrey K. Lazo: Communicating uncertainty in weather forecasts: A survey of the U.S. public, Weather and Forecasting 23 (2008)

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 32

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow (at least 0.1 mm at the weather station)

◮ It will rain tomorrow in 30% of the Hull region.

× 16%

◮ It will rain tomorrow for 30% of the time.

× 10%

◮ 30% of weather forecasters believe it will rain tomorrow.

× 22%

◮ It will rain on 30% of the days like tomorrow.

19%

Rebecca E. Morss, Julie L. Demuth, and Jeffrey K. Lazo: Communicating uncertainty in weather forecasts: A survey of the U.S. public, Weather and Forecasting 23 (2008)

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 33

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow (at least 0.1 mm at the weather station)

◮ It will rain tomorrow in 30% of the Hull region.

× 16%

◮ It will rain tomorrow for 30% of the time.

× 10%

◮ 30% of weather forecasters believe it will rain tomorrow.

× 22%

◮ It will rain on 30% of the days like tomorrow.

() 19%

Rebecca E. Morss, Julie L. Demuth, and Jeffrey K. Lazo: Communicating uncertainty in weather forecasts: A survey of the U.S. public, Weather and Forecasting 23 (2008)

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 34

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow (at least 0.1 mm at the weather station)

◮ It will rain tomorrow in 30% of the Hull region.

×

◮ It will rain tomorrow for 30% of the time.

×

◮ 30% of weather forecasters believe it will rain tomorrow.

×

◮ It will rain on 30% of the days like tomorrow.

()

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 35

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow (at least 0.1 mm at the weather station)

◮ It will rain tomorrow in 30% of the Hull region.

×

◮ It will rain tomorrow for 30% of the time.

×

◮ 30% of weather forecasters believe it will rain tomorrow.

×

◮ It will rain on 30% of the days like tomorrow.

()

◮ It will rain on 30% of the days for which

the weather forecast says 30% probability of rain.

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 36

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow (at least 0.1 mm at the weather station)

◮ It will rain tomorrow in 30% of the Hull region.

×

◮ It will rain tomorrow for 30% of the time.

×

◮ 30% of weather forecasters believe it will rain tomorrow.

×

◮ It will rain on 30% of the days like tomorrow.

()

◮ It will rain on 30% of the days for which

the weather forecast says 30% probability of rain. ()

Marco Cattaneo @ University of Hull What is probability? 4/7

SLIDE 37

example: probability of rain

Hull weather forecast: 30% probability of rain for tomorrow (at least 0.1 mm at the weather station)

◮ It will rain tomorrow in 30% of the Hull region.

×

◮ It will rain tomorrow for 30% of the time.

×

◮ 30% of weather forecasters believe it will rain tomorrow.

×

◮ It will rain on 30% of the days like tomorrow.

()

◮ It will rain on approximately 30% of the days for which

the weather forecast says 30% probability of rain.

• Marco Cattaneo @ University of Hull

What is probability? 4/7

SLIDE 38

games of chance

Marco Cattaneo @ University of Hull What is probability? 5/7

SLIDE 39

games of chance

probability of an event = number of favorable outcomes number of possible outcomes

Marco Cattaneo @ University of Hull What is probability? 5/7

SLIDE 40

games of chance

probability of an event = number of favorable outcomes number of possible outcomes

◮ Girolamo Cardano (Pavia 1501 – Rome 1576):

Liber de ludo aleae (1663, written around 1564)

Marco Cattaneo @ University of Hull What is probability? 5/7

SLIDE 41

games of chance

probability of an event = number of favorable outcomes number of possible outcomes

◮ Girolamo Cardano (Pavia 1501 – Rome 1576):

Liber de ludo aleae (1663, written around 1564)

◮ Blaise Pascal (Clermont-Ferrand 1623 – Paris 1662) and

Marco Cattaneo @ University of Hull What is probability? 5/7

SLIDE 42

games of chance

probability of an event = number of favorable outcomes number of possible outcomes

◮ Girolamo Cardano (Pavia 1501 – Rome 1576):

Liber de ludo aleae (1663, written around 1564)

◮ Blaise Pascal (Clermont-Ferrand 1623 – Paris 1662) and

Pierre de Fermat (Beaumont-de-Lomagne 1601/1607 – Castres 1665): correspondence on the problem of division of the stakes (1654)

Marco Cattaneo @ University of Hull What is probability? 5/7

SLIDE 43

games of chance

probability of an event = number of favorable outcomes number of possible outcomes

◮ Girolamo Cardano (Pavia 1501 – Rome 1576):

Liber de ludo aleae (1663, written around 1564)

◮ Blaise Pascal (Clermont-Ferrand 1623 – Paris 1662) and

Pierre de Fermat (Beaumont-de-Lomagne 1601/1607 – Castres 1665): correspondence on the problem of division of the stakes (1654)

◮ Christiaan Huygens (The Hague 1629 – The Hague 1695):

De ratiociniis in ludo aleae (1657)

Marco Cattaneo @ University of Hull What is probability? 5/7

SLIDE 44

PS¥CHOMETRIKA--VOL. 3{},

• NO. 1

MARCH, 1971

BIAS AND RUNS IN DICE THROWING AND RECORDING: A FEW MILLION THROWS* GUDMUND R. IVERSEN, WILLARD H. LONGCORt, FREDERICK MOSTELLER, JOHN P. GILBERT, AND CLEO YOUTZ

An experimenter threw individually 219 different dice of four different brands and recorded even and odd outcomes for one block of 20,000 trials for each die---4,380,000 throws in all. The resulting data on runs offer a basis for comparing the observed properties of such a physical randomizing process with theory and with simulations based on pseudo-random numbers and RAND Corporation random numbers. Although generally the results are close to those forecast by theory, some notable exceptions raise questions about the surprise value that should be associated with occurrences two standard deviations from the mean. These data suggest that the usual significance level may well actually be running from 7 to 15 percent instead

• f the theoretical 5 percent.

The data base is the largest of its kind. A set generated by one brand of dice contains 2,000,000 bits and is the first handmade empirical data of such size to fail to show a significant departure from ideal theory in either location

• r scale.
• 1. Introduction

How well do the laws of chance actually work? When a die is repeatedly thrown and its outcomes recorded, do imperfections in the die, in the throwing, in the perception of the outcome, and in recording appear? What sorts of deviations from chance do we find? Weldon's dice data [Fry, 1965] and Kerrieh's coin tossing monograph [Kerrieh, 1946] both give us some experience with large bodies of data pro- duced by humanly run physical randomizing devices whose idealized prob- abilities and properties are known to a good approximation. In a sense, such experiments are controls on other experiments where probability plays an important role. For example, such dice and coin experiments give us an idea of how seriously we should take small departures from mathematically predicted results in investigations where we search for small departures from a standard. They do this by showing the sizes and kinds of departures

• bserved in an experiment with no planned human or material effects. They

are placebo experiments. If one does not believe in extra-sensory perception, then many ESP investigations also would be iudged to qualify, but if one * The analysis was facilitated by a National Science Foundation grant GS-341 and and its continuation GS-2044X. It forms part of a larger study of data analysis. t Mr. Longcor is from Waukegan, Illinois; the other authors are from Harvard

• University. Dr. Iversen has moved to the University of Michigan.

1

SLIDE 45

PS¥CHOMETRIKA--VOL. 3{},

• NO. 1

MARCH, 1971

BIAS AND RUNS IN DICE THROWING AND RECORDING: A FEW MILLION THROWS* GUDMUND R. IVERSEN, WILLARD H. LONGCORt, FREDERICK MOSTELLER, JOHN P. GILBERT, AND CLEO YOUTZ

An experimenter threw individually 219 different dice of four different brands and recorded even and odd outcomes for one block of 20,000 trials for each die---4,380,000 throws in all. The resulting data on runs offer a basis for comparing the observed properties of such a physical randomizing process with theory and with simulations based on pseudo-random numbers and RAND Corporation random numbers. Although generally the results are close to those forecast by theory, some notable exceptions raise questions about the surprise value that should be associated with occurrences two standard deviations from the mean. These data suggest that the usual significance level may well actually be running from 7 to 15 percent instead

• f the theoretical 5 percent.

The data base is the largest of its kind. A set generated by one brand of dice contains 2,000,000 bits and is the first handmade empirical data of such size to fail to show a significant departure from ideal theory in either location

• r scale.
• 1. Introduction

How well do the laws of chance actually work? When a die is repeatedly thrown and its outcomes recorded, do imperfections in the die, in the throwing, in the perception of the outcome, and in recording appear? What sorts of deviations from chance do we find? Weldon's dice data [Fry, 1965] and Kerrieh's coin tossing monograph [Kerrieh, 1946] both give us some experience with large bodies of data pro- duced by humanly run physical randomizing devices whose idealized prob- abilities and properties are known to a good approximation. In a sense, such experiments are controls on other experiments where probability plays an important role. For example, such dice and coin experiments give us an idea of how seriously we should take small departures from mathematically predicted results in investigations where we search for small departures from a standard. They do this by showing the sizes and kinds of departures

• bserved in an experiment with no planned human or material effects. They

are placebo experiments. If one does not believe in extra-sensory perception, then many ESP investigations also would be iudged to qualify, but if one * The analysis was facilitated by a National Science Foundation grant GS-341 and and its continuation GS-2044X. It forms part of a larger study of data analysis. t Mr. Longcor is from Waukegan, Illinois; the other authors are from Harvard

• University. Dr. Iversen has moved to the University of Michigan.

1

SLIDE 46

law of large numbers

JACOB1 BERNOULLI,

P S ~ E ~ K

B a a . & utriufque Societ. Reg. Scientiar.

Gall, & PruK Sodal. ~ ~ A T H E M . $ T I C I CEEEBERRIMI,

ARS CONJECTANDI,

OPUS POSTHUMUM*

T R A C T A ? ' U , S

DE

SERIEBUS INFINITIS,

B A S I L E R ,

Impenfs THURNISIOKUM, Fratnun.

cI3 b c c X I 11.

Marco Cattaneo @ University of Hull What is probability? 6/7

SLIDE 47

law of large numbers

probability of an event = long-run relative frequency of its occurrence

JACOB1 BERNOULLI,

P S ~ E ~ K

B a a . & utriufque Societ. Reg. Scientiar.

Gall, & PruK Sodal. ~ ~ A T H E M . $ T I C I CEEEBERRIMI,

ARS CONJECTANDI,

OPUS POSTHUMUM*

T R A C T A ? ' U , S

DE

SERIEBUS INFINITIS,

B A S I L E R ,

Impenfs THURNISIOKUM, Fratnun.

cI3 b c c X I 11.

Marco Cattaneo @ University of Hull What is probability? 6/7

SLIDE 48

law of large numbers

probability of an event = long-run relative frequency of its occurrence Jacob Bernoulli (Basel 1655 – Basel 1705): Ars conjectandi (1713)

JACOB1 BERNOULLI,

P S ~ E ~ K

B a a . & utriufque Societ. Reg. Scientiar.

Gall, & PruK Sodal. ~ ~ A T H E M . $ T I C I CEEEBERRIMI,

ARS CONJECTANDI,

OPUS POSTHUMUM*

T R A C T A ? ' U , S

DE

SERIEBUS INFINITIS,

B A S I L E R ,

Impenfs THURNISIOKUM, Fratnun.

cI3 b c c X I 11.

Marco Cattaneo @ University of Hull What is probability? 6/7

SLIDE 49

JACOB1 BERNOULLI,

P S ~ E ~ K

B a a . & utriufque Societ. Reg. Scientiar.

Gall, & PruK Sodal.

~ ~ A T H E M . $ T I C I

CEEEBERRIMI,

ARS CONJECTANDI,

OPUS POSTHUMUM*

T R A C T A ? ' U , S

DE

SERIEBUS INFINITIS,

B A S I L E R ,

Impenfs THURNISIOKUM, Fratnun.

cI3 b c c X I

11.

SLIDE 50

SLIDE 51

SLIDE 52

• ••,. 'on

H 7

~

tho maximum M and du: bound LwiU =<e<I mo", chan c(, _ 1) rima

,I....me oumbe. of term> ..ortiog from ,he b'F' f<Tm' oul>ide ,hi. bound.

Simil""Iy. 'h"" will "",«<I _ « m.n <oi""" ,Iu, many""'''''' uk<n . _ 1 ,;m . 1b<t<fo«. <Y<f1 mol"< o[".i"",ly. th<y will ae«<! "'0« ,h.n <'ima . 11 'ho te,....

"""ide ,be bound 1., uf which ,h... ... unly J - I 'imes .. many mo...

For the urm. ru ,h. ,igh,. 1 proc.ed in ,h• ..,.,. w.y. I ,ok. ,h. ",io (, . 1)1 . < (" . 1')1(" _ ,), I «t (, .

1 )

~1 .~

~

c(, _ I), and I find th.. '" O!: 1

,,&

[dt - IJ jljlug (• • I) _ log 'I- Nat. in tho «tie> of fr.a<tioM (~

"

• n

t J I

( ~ " - '" .,J

. ("" • n' _ n l(o" _ "' . 2J) . (n" • n' _ 2<11(0" _ '" • ,i,) •. • (n" • <II"...

which .ignify ,h. ra' io MIA. I "'P!""" 'M' ,he frac'ion in 'he ""h po.i'ion. rwnely (0'"

", _ Of"

,)I

(~"

_ It<. ",,) , i> "'lu.al ' 0 ( , . l)J.

. and from ,hi. I

find ,Iu, n .. '" • ("" _ .)1 (, , 1). and hrn« th.' 0' • "" . (•.". _ ,,)1(•• I). Thi> h.ving beoen <1<>.... it i• •imibtly shuwn, .. hef"... ,M,. when ,he Mnumial

.. , i. ...1<m '0 ,h" 1""'-'. i" l>Wlimum term M ..««It ,he bound A mol< ,h.n

d . - l) ,im... Con>«jom,ly al",. all .he t<rm. included ~

,he m""imum

Mand ,h. bound /\ <Il«e<! b)' mu.. than rtimes all,h. ,.rm. ou"id. ,hi, bound.

• fwhidl me..... only , _ 1 'imes mOt<,Thus finally. io 'he end, .... conclude

,h., when the binomial , • • i> raiood '0 ,he pow« of which ,be ind.. i, "'Iu>I '0

'ho 1.l'lJ<' of 'h<!e twO qlWl,itid.. [2l6J"',. (",,'- 11)1

(.. Ih nd "', • (",n - ".jJ

(• • I). wn <h••um of [he [eo"" indud<d b.""""n the twO bound. L and A

=eetb by m""h mo", ,h'n ( ,im.. ,h. ,U

rn of ,h. ,otnu beyond ,h. bound.

• n bo,h .id... Th' refore a hni[. po...er h.. beeo fou nd [ha, h.. ,h. d..irN

PlO('<"Y, Q.E.D. p,.;""jvf "",,.,in'.o, Finally

. th«e f"lk>w> ,he proposi'ion for ,he..1, of which all [hi. h.. hoeo <aid. hu, wfIa« demon"'atiotl un no... be 1Ii,..n with

• nly.he appl""t;"" of'ho for<g<>ingInn",..,To ",oid ,edious circumlo<ution.

I will all "" c»<s io which . t<tUin <Ven[ con fupren Im",J or fmill, I will

all ,tmll ""'" ca>e$ in which ,h. eYf1I' can "'" fuppen.l will also call..pen-

mClt l>frct."" orImiJe in which one ofthe fertile ca>e$ is dilCO\'erN '0 occur; and I will call 1

• •mftr1<"" 0' ,utiii' ,h"", in which 0'''' of"" ""ii, .......
• o..:rv>N ro lupp<n. l.<'r ,he "u",ber of f..,ik ""'" and [ho numbe, of " ..ik

'""" hav. eualy '" apptm.ima[dy tho ruio tI

~ rnd let ,he oumber of~i

l ......

to.ll ,he,""" he in ,he ratio tI(,. ,) or tI,. wfIich ,atio ;, bounded by ,he lim·

ill (" 1)1' and (t _ I ) I ~

It i> '0 b< ."""'n ,fu[.., many «p"rim..." un be takrn

'fu' i' become> any ~

v< n

number of 'imo (..y n imes) mon: likely l....,:,.-..;J·

ilU] ,fu, ,h. number off."il. oh.ef>oatio'" will f:oIl berv....n ,h... bound. ,han

"u",ide ,hem. ,h.. i•. ,hat ,he ra'io or,ho "umber <>f(""il< to 'ho number of all [he oo..rv"ion. will h.v< • ,,[;., ,hat is "ei[h" mon: [han (t • 1)1' nor .... ,han (,_ 1)1'.

• Copy. ghtoo m

al

SLIDE 53

conclusion

“Il est remarquable qu’une science qui a commenc´ e par la consid´ eration des jeux, se soit ´ elev´ ee aux plus importans objets des connaissances humaines.”

Pierre-Simon de Laplace: Essai philosophique sur les probabilit´ es (1814)

[“It is remarkable that a science which began with the consideration of games of chance should have raised itself to the most important objects of human knowledge.”]

Marco Cattaneo @ University of Hull What is probability? 7/7

SLIDE 54