\documentclass[12pt]{article}

\begin{document}

 

\noindent{\bf Blaise PASCAL}\\
b. 19 June 1623 - d. 19 August 1662
\vspace{.5 cm}

\noindent
{\bf Summary} Pascal introduced the concept of mathematical expectation and 
used it recursively to obtain a solution to the Problem of Points which was the 
catalyst that enabled probability theory to develop beyond mere combinatorial 
enumeration.
\vspace{.5cm}

Blaise Pascal was born in Clermont, France. In 1631 his father
Etienne Pascal (himself an able mathematician who gave his name to the
'{\it lima\c{c}on} of Pascal') moved his family to Paris
in order to secure his son a better education, and in 1635 was one of
the founders of Marin Mersenne's 'Academy', the finest exchange of mathematical 
information in Europe at the time. To this informal academy he introduced his 
son at the age of fourteen, and Blaise immediately put his new source of 
knowledge to good use, producing (at the age of sixteen) his 
{\it Essay pour les coniques}, a single printed sheet 
enunciating Pascal's Theorem, that the opposite sides of a hexagon inscribed 
in a conic intersect in three collinear points. 

At the age of eighteen the younger Pascal turned his attention to constructing 
a calculating machine (to help his father in his calculations) and within a 
few years he had built and sold fifty of them. Some still exist. 
In 1646 he started work on hydrostatics, 
determining the weight of air experimentally and writing on the vacuum (leading 
ultimately to the choice of 'Pascal' as the name for the S.I. unit of pressure). 

In 1654 Pascal returned to mathematics, extending his early work on conics in a 
manuscript which was never printed and which does not now exist, though it was 
seen by Leibniz. In the same year he entered into correspondence with Pierre de
Fermat (q.v.) of Toulouse about some problems in calculating the odds 
in games of chance which led him to write the {\it Trait\'e du
triangle arithm\'etique, avec quelques autres petits traitez
sur la mesme mati\`ere}, probably
in August of that year. Not published until 1665, this work, and the 
correspondence 
itself which was published in 1679, is the basis of Pascal's reputation in 
probability theory as the originator of the concept of expectation and its use 
recursively to solve the 'Problem of Points', as well as the justification for 
calling the arithmetical triangle 'Pascal's triangle'. His advances, 
considered to be the foundation of modern probability theory, are described in 
detail below. 

Later in 1654 Pascal underwent a religious experience as a result of 
which he 
almost entirely abandoned his scientific work and devoted his remaining 
years to 
writing his {\it Lettres provinciales}, written in 1656 
and 1657 in defence of Antoine Arnauld, and his {\it Pens\'ees}, 
drafts for an {\it Apologie de la Religion Chr\'etienne}, 
which include his famous 'wager' (see below). In 1658-59 Pascal briefly 
returned 
to mathematics, writing on the curve known as the cycloid, but his 
final input into 
the development of probability theory arises through his presumed 
contribution to 
{\it La logique, ou l'art de penser} by Antoine Arnauld 
and Pierre Nicole, published in 1662. This classic philosophical 
treatise is often 
called the {\it Port-Royal Logic} through the association 
of its authors, and Pascal himself, with the Port-Royal Abbey, a centre 
of the 
Jansenist movement within the Catholic church, into whose convent 
Pascal's sister 
Jacqueline had been received in 1652. 

Pascal died in Paris, in good standing with the
Church and is buried in the church of St Etienne 
du Mont. 

The {\it Trait\'e du triangle arithm\'etique} itself is 36 pages 
long (setting aside {\it quelques autres petits traitez sur 
la mesme mati\`ere}) and consists of two parts. The first carries 
the title by which the 
whole is usually known, in English translation {\it A Treatise 
on the Arithmetical Triangle}, and is an account of the arithmetical 
triangle as a 
piece of pure mathematics. The second part {\it Uses of the 
Arithmetical Triangle} consists of four sections: 
(1){\it Use in the theory of figurate numbers}, 
(2){\it Use in the theory of combinations},
(3){\it Use in dividing the stakes in games of chance}, 
(4){\it Use in finding the powers of binomial expressions}. 

Pascal opens the first part by defining an unbounded rectangular array 
like a matrix in 
which 'The number in each cell is equal to that in the preceding cell in 
in the same row', and he considers the special case in which the
cells of the first row and column each contain 1. Symbolically,
he has defined $\{f_{i,j}\}$ where
$$f_{i,j} = f_{i-1,j} + f_{i,j-1}, i,j= 2,3,4,..., f_{i,1}=
f_{1,j} = 1, i,j = 1,2,3,...$$

The rest of Part I is devoted to a demonstration of nineteen corollaries 
flowing from this definition, and concludes with a 'problem'. The corollaries
include all the common relations among the {\it binomial 
coefficients} (as the entries of the triangle are now universally called), 
none of which was new. Pascal proves the twelfth corollary, 
$(j-1)f_{i,j} = if_{i+1,j-1}$ in our notation, by explicit use of
mathematical induction. The `problem' is to find $f_{i,j}$
as a function of $i$ and 
$j$, which Pascal does by applying the twelfth corollary recursively. Part I of 
the {\it Treatise} thus amounts to a systematic development of all the 
main results then
known about the properties of the numbers in the arithmetical triangle.

In Part II Pascal turns to the applications of these numbers. The numbers 
thus defined have three different interpretations, each of great
antiquity (to which he does not, however, refer).
The successive rows of the triangle 
define the {\it figurate numbers} which have their roots in
Pythagorean arithmetic. Pascal treats these in section (1),

The second interpretation is as {\it binomial numbers}, 
the coefficients of a binomial expansion, which are arrayed in the 
successive 
diagonals, their identity with the figurate numbers having been recognized in 
Persia and China in the eleventh century and in Europe in the sixteenth. The 
above definition of $f_{i,j}$ is obvious on considering the expansion of 
both sides of 
$$(x+y)^n = (x+y) (x+y)^{n-1}.$$ 
The fact that the coefficient of $x^{r} y^{n-r}$ in the expansion of 
$(x+y)^n$ may be expressed as 
$$ \frac{n(n-1)(n-2)...(n-r+1)}{1.2.3...r} =  {n \choose r}$$ 
was known to the Arabs in the thirteenth century and to the Renaissance 
mathematician 
Cardano in 1570. It provides a closed form for $f_{i,j}$, with 
$n =  i + j -2$ and $r = i -1$. 
Pascal treats the binomial interpretation in section (4). 

The third interpretation is as a {\it combinatorial number}, for the
number of combinations of $n$ different things taken $r$ at a time,
$^{\fontshape{it}\selectfont  {} n}$C$_{\fontshape{it}\selectfont  r}$
is equal to
${ n \choose r}$,
a result known in India in the ninth century, to Hebrew writers in the 
fourteenth century, and to Cardano in 1550. Pascal deals with
that interpretation in section (2), giving a 
novel demonstration of the combinatorial version of the basic 
addition relation 
$^{\fontshape{it}\selectfont  n+}$$^{  1}${\fontshape{it}\selectfont C}$_{\fontshape{it}\selectfont  r+1}${} 
= $^{\fontshape{it}\selectfont  {} n}$C$_{\fontshape{it}\selectfont  r}${} + 
$^{\fontshape{it}\selectfont  {} n}${\fontshape{it}\selectfont C}$_{\fontshape{it}\selectfont  r+}$$_{  1}$, 
for, considering any particular one of the $n+1$ things, 
$^{\fontshape{it}\selectfont  n}$C$_{\fontshape{it}\selectfont  r}${} 
gives the number of combinations that include it and 
$^{\fontshape{it}\selectfont  n}${\fontshape{it}\selectfont C}$_{\fontshape{it}\selectfont  r+}$$_{  1}${} 
the number that exclude it.

In section (3) 
Pascal breaks new ground, and this section, taken together with 
his correspondence with Fermat, is the basis of his reputation as the father 
of probability theory. The ``problem of points'' which he discusses, also 
known simply as the ``division problem'', involves determining how the total 
stake should be divided in the event of a game of chance being terminated 
prematurely. Suppose two players $X$ and 
$Y$ stake equal money on being the first to win 
$n$ points in a game in which the winner of each 
point is decided by the toss of a fair coin. If such a game is interrupted 
when $X$ still lacks $x$
points and $Y$ lacks $y$, 
how should the total stake be divided between them? In the middle of the 
sixteenth century Tartaglia famously concluded ``the resolution of such a 
question is judicial rather than mathematical, so that in whatever way the 
division is made there will be cause for litigation''. A century later, in 
the summer of 1654, the correct solution was derived by three different 
arguments during the correspondence between Pascal and Fermat. One of the 
methods, advanced by Pascal in his letter of July 29, involves computing 
expectations recursively. 

The key development was his understanding that the value of a gamble is equal 
to its mathematical expectation computed as the average of the values of each 
of two equally-probable outcomes, and that this precise definition of value 
lends itself to recursive computation because the value of a gamble that one 
is certain to win is undoubtedly the total stake itself. Thus if the 
probabilities of winning $a$ or $b$ 
units are each one half, the expectation is $(a + b)$
units, which is then the value of the gamble. 
In {\it Ars conjectandi} (1713) James Bernoulli (q.v.) called 
this ``the fundamental principle of the whole art''. Pascal has invented the 
concept of ``expected value'', that is the probability of a win multiplied by 
its value, and understood that it is an exact mathematical concept which can 
be manipulated. 

In the letter to Fermat, Pascal develops the recursive argument applied to 
expected values in order to find the correct division of the stake-money, 
and thus computes the ``value'' of each successive throw. In the 
{\it Trait\'e} the same idea is more formally expressed, 
and in particular Pascal gives as a principle the value of the expectation 
when the chances are equal. He remarks at one point that ``the division has 
to be proportional to the chances'', but in the solution to the problem when 
the players have equal chances the question of computing an expectation for 
unequal chances does not arise; that extension was first formally made by 
Huygens (q.v.) in his {\it De ratiociniis in ludo aleae} of 
1657. It is known that when Huygens spent July-September 1655 in Paris he 
had the opportunity to discuss Pascal's work on probability problems with 
Roberval, and presumably learnt of the concept of mathematical expectation then. 

The easiest way to understand how Pascal used expectation and recursion to 
solve the problem of points is to visualise the event-tree of possible further 
results. Each bifurcation corresponds to a toss, one branch for 
$X$ winning it and the other for 
$Y$, and successive bifurcations must lead eventually 
to tips corresponding to the whole game being won by either 
$X$ or $Y$. Considering 
now the expectation of $X$ (say), each tip can be 
labelled with his expectation, either $S$ (the total 
stake) or 0, as the case may be. Applying now Pascal's expectation rule for 
equal 
chances, each bifurcation can have an expectation associated with it, working 
recursively down the tree from the tips to the root, at which point we find the 
solution to the problem. If $E(x,y)$ 
be the expectation of player $X$ 
when he lacks $x$ points and $Y$
lacks $y$, then the recursion is 
$$ E(x,y) = E(x-1,y) + E(x,y-1).$$

By these methods, and his knowledge of the arithmetical triangle, Pascal was 
able to demonstrate how the stake should be divided between the players 
according 
to the partial sums of the binomial coefficients, a result which he had already 
obtained by enumeration, for both he and Fermat had realised that the actual 
game 
of uncertain length could be embedded in a game of fixed length to which the 
binomial distribution (as yet undiscovered) could then be applied. Further 
details of their reasoning may be found in the references below. 

Three further contributions of Pascal to the development of 
probability theory
may be noted. In {\it Pens\'ees} he used his concept
of expectation to argue that one should bet on the existence of God because, 
however small the probability of His existence, the value
of eternal salvation
if He does exist is infinite, so that the expected value of assuming that He 
does exist far exceeds that of assuming that He does not. Subsequent writers 
have regarded this, ``Pascal's wager'', as the origin of decision theory. 
Secondly, in 1656 Pascal posed Fermat a problem which became known as the 
``gambler's ruin'' and which played a central role in the development of 
probability theory as the first example of a problem about the duration of 
play. Finally, he greatly influenced the probability arguments in the 
{\fontshape{it}\selectfont Port-Royal Logic}, in the last chapter of which 
there is a clear understanding of the importance of judging an action not only 
by the possible gain or loss, but also by the probability of each of these. 
One must ``consider mathematically the magnitudes when these things are 
multiplied together''. This is the first occasion in which the word ``
probability" is used (in French, of course) in its modern sense.
\vspace{.5 cm}

\begin{thebibliography}{3}

\bibitem{1}  Edwards, A.W.F. (1987). {\it Pascal's Arithmetical Triangle},
Griffin, London, and Oxford University Press, New York, 2nd ed. 2000,
Johns Hopkins University Press, Baltimore. Contains as appendices
Pascal and the Problem of Points {\it International Statistical Review},
{\bf 50}, 259-266 (1982) and Pascal's Problem: The "Gambler's Ruin",
{\it International Statistical Review}, {\bf 51}, 73-79 (1983).

\bibitem{2}  Mesnard, J. (1970). {\it Oeuvres comp\'etes de Blaise Pascal},
deuxi\`eme partie volume I: Oeuvres diverses. Decl\'ee de Brouwer, Bruges.

\bibitem{3}  Hald, A. (1990)  {\it A History of Probability and Statistics 
and their Applications before 1750}. Wiley, New York.

\bibitem{4}  Pascal, B. (1955, 1963) {\it Great Books of the Western World 33: 
The Provincial Letters, 
Pens\'ees,} Scientific Treatises, by Blaise Pascal. Encyclopadia Britannica, 
Chicago. 
Contains the best English translations of the Pascal-Fermat correspondence 
and the {\it  
Trait\'e du triangle arithm\'etique}.
\vspace{1 cm}

\hfill{A.W.F. Edwards}   


\end{thebibliography}

\end{document}

