# Chapter IX The Development of Probability Theory Pascal_ Bernoulli by dffhrtcv3

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```									   Chapter IX

The Development of
Probability Theory:
Pascal, Bernoulli,
Bernoulli, and Laplace
Origins of Probability Theory
– The solution of wagering problems connected
with games of chance
– The process of statistical data for such matters
as insurance rates and mortality tables
• 14th Century
– Needed an estimate on mortality rates
because towns were losing money to people
who died early
John Graunt (1620-1674)
• Wrote Natural and Political Observations
Made upon the Bills of Mortality
• Bills of Mortality
– weekly and yearly returns on the number of
burials in several London perishes
– Counted deaths because of the plague
– Did not distinguish age, only sex and cause
(disease or accident)
• Used statistics for the 57 years from 1604-
1661
• Came up with table on pg 404
Christian Huygens
(1629-1695)
• Used Graunt’s figures to determine when
people would die
• Estimated his death at 55 and his
brother’s at 56½
• Huygens wrote De Ratiociniis in Ludo
Aleae (On Reasoning in Games of
Chance)1st text on probablity theory
published
Gaming
• Many believed nothing was random, a
divine being (God or gods) influenced
earthly events
• They could interfere with the throwing of
dice.
• Gambling was know in Egypt 3500BC with
a game called “Hounds and Jackals.”
• Games were used during the famine
– One day of pure games to keep their minds
off food
– One day for food
Restrictions
• Greeks and Romans issued laws
forbidding gaming except certain seasons
because of popularity
• Forbidden by Jews under penalty of death
because the gamblers always expects to
win, and therefore get something for
nothing. Rabbis considered this immoral
and akin to robbery
Restrictions
• Christian Churches banned gaming
because of the accompanying vices of
dringking and swearing
the rank of knight could gamble and
those who could, weren’t allowed to lose
more that 20 shillings in a 24 hour period.
Cards
• Exact origin is murky
• Credited to Egyptians, Chinese, and
Indians (~1300’s)
• 1st were hand painted
• Gutenberg made the 78-card tarot deck
• Original portraits of actual people David
Alexander the Great (clubs), Julius Cæsar
(diamonds)
Cardan
• Wrote Liber de Ludo Aleae – 1st reasoned
considerations relating to games of
chance
• Stated that the probability of a particular
outcome was the sum of the possible
ways of achieving that outcome divided
by the totality of possible outcomes of an
event
Cardan
• Found a transition from empiricism to
theoretical concept of a fair die
• Probably became the real father of
modern probability theory
Blaise Pascal (1623-1662)
• Closely linked with Fermat as one of the
joint founders of probability theory
• A gifted writer, religious philosopher,
creative mathematician, and experimental
physicist
• Theologian
History of Pascal
• Raised without ever going to school or
studying at a university
• Taught exclusively by his father
• Kept from studying math until age 15
– All math books were locked up and the
subject was forbidden
History of Pascal
• At 12, wanted to know what geometry
was and worked on it alone drawing
charcoal figures on floor tiles
• His father caught him trying to prove his
guess that the sum of the angles of a
triangle is two right angles (Euclid’s Prop
32)
• He than was given a copy of Euclid’s
Elements and mastered them without
assistance
Essay pour les coniques
• Pascal’s mystic hexagon
• In this essay, Pascal deduced no fewer
that 400 propositions on conic sections
from the mystic hexagon theorem
Calculator
• Saw his father with lots of addition work
and thought of a contraption to add and
multiply eight figures
• Named the Pascaline (like calculators of
1400’s)
• Too expensive to buy so it was more a
curiosity than a useful device
Cycloid -1658
• Cycloid – the curve traced out by a point
on the circumference of a wheel, as the
wheel rolls along a straight line.
http://archives.math.utk.edu/visual.calculus/0/parametric.5/

• Pascal found out how to get area under
one arch and the volume of the solid
obtained revolving the curve about the
base line
Cycloid -1658
• The work of Pascal on the cycloid had one
by product of surprising importance
– Inspired Leibniz in his invention of he
differential and integral calculus
• One of Pascal’s letters involved certain
calculations that resembled the evaluation
of the definition of the sine function
De Méré
• De Méré made a precarious living at cards
and dive
• One of De Méré’s gaming problems gave
Pascal new insight
– Page 417
Pascal’s Triangle
• The arithmetic triangle, now generally known as
Pascal’s triangle, is an infinite numerical table in
“triangular form,” where the nth row of the
triangle lists the successive coefficients in the
binomial expansion of x  y    n

• Was thought of many times before Pascal
–   Chu Shih-Chen in 1303
–   Chia Hsien in 1050
–   Omar Khayyam
–   Al-Tusi
• Printed by Peter Apian in 1527
Pascal’s Triangle
• Linked to Pascal because he was the first to
make any sort of systematic study of the relation
it exhibited.
• Wrote Traité du Triangle Arithmétique – an
exposition of the properties and relations
between the binomial coefficients
Recursive formula:
x  y      x   x y   x y   
n-1       n- 2 2
xy   y
n-1
n     n   n   n          n              n             n    n
0       1          2              n1           n
– Stated in this symbolism, the property of the
numbers  n  on which the triangle is based
      
r

n      n        n1
r 1   r        r
– This rule does not give a formula for the
binomial coefficients, but tells us only how to
build them from 2 numbers on the previous
base.
Explicit formula for      
n
r


n
r   
n!
r!n  r !
Consequences
• Pascal listed 19 consequences of the
binomial coefficients that could be
discovered for the triangle.
• Tells us that each number in the triangle is
the sum of entries in the preceding
diagonal, beginning with the entry above
the given number
Consequence XII

  nr
n
r 1
  r 1
n
r
• Proof on pg 427
• Important because it’s proof involves the
1st explicit formulation of the
demonstrative procedure known as
induction
Induction
• The proof of a proposition P(n) by
mathematical induction consists of
showing that:
1. The statement P(n0) is true for some
particular integer n0.
2. The assumed truth of P(k) implies the truth
of P(k+1)
Activity:
Prove sum of cubes by induction!
 nn  1 n n  1
n               2    2       2

j  2   4
j 1
3

         
Christian Huygens
(1629-1695)
• Taught by tutors at home
• Went to University of Leiden
• By 1666, he was known as a physicist,
astronomer, and mathematician
• Wrote Traité de Lumiére describing his radically
new wave theory of light
• Invented pendulum clock
Christian Huygens
(1629-1695)
• Came up with formula T  2
l
g
relating to oscillating period T of a simple
pendulum undergoing small swings with it’s
length l. This afforded a practical way of
measuring g, the acceleration due to gravity.
• Wrote De Ratiociniis in Ludo Aleae (On
Mathematical Probability)
• Most important discovery ws the important
concept of mathematical expectation (or as he
called it the value of chance of winning a game)
James (Jacques, Jacob)
Bernouilli
• Became a minister and studied math and
astronomy against his father’s wishes.
• He was the first to achieve full understanding of
Leibniz’s differential calculus
• He taught these to his younger brother John.
James (Jacques, Jacob)
Bernouilli
• Bernouilli’s contributions to mathematics is Ars
Conjectandi (the Art of Conjecture)
– First part: A reproduction of Huygen’s De Ratiociniis
with commentary.
– Second part: all the standard results on permutations
and combinations.
– Third part: Consists of 24 problems relating to the
various games of chance that were popular in
Bernouilli’s day.
– Fourth part (Most important): Bernouilli’s theorem:
if p is the probability of an event, if k is the actual
number of times the event occurs in n trials, if e > 0 is
an arbitrarily small number, and if P is the probability
k
that the inequality n  p  e is satisfied, then P
increases to 1 as n grows with out bound.
John (Jean, Johann)
Bernouilli
• Studied medicine and was tutored by his
brother in mathematics privately
• Followed in James’s footsteps by leading
exponents of the calculus
• Antagonism devolved and the two brothers
became rivals
• Came up with a formula for a curve along which
a body only affected by gravity would fall with
constant velocity:

dy b y  a  dx a
2     2        3
John (Jean, Johann)
Bernouilli
• John Bernouilli tutored L’Hospital
• L’Hospital published various discoveries given to
him by Bernouilli
• Bernouilli fought back after L’Hospital died
• L’Hospital’s work was proven to be Bernouilli’s
De Moivre
• Wrote Doctrine of Chances: or a Method of
Calculating the Probability of Events in Play
(1718)
• Supported himself by solving problems
proposed to him by wealth patrons who wanted
to know what stakes to offer in games of
chance.
• His book contained numerous problems on
throwing dice and other probabilities
De Moivre
• Became lethargic and predicted his own death
– He was sleeping 20 hours a day and finding he was
sleeping a quarter of an hour longer than on the
preceding day
– He calculated he would die in his sleep on the very
day in which he slept up to the limit of 24 hours.
– He died at the age of 87; the cause of his death was
recorded as “somnolence”
(i.e. sleepiness)
Pierre Laplace (1749 – 1827)
• Educated in school between ages of 7 and 16
• Went to University of Caen at 16
• Intended study theology then decided his true
vocation lay in mathematics.
• Greatest achievement was Traité de Mécanique
Céleste
– Which is designed to solve the great and mechanical
problems of the solar system and to bring theory to
coincide so closely with observation that empirical
equations should no longer be needed.
Pierre Laplace (1749 – 1827)
• Frequently unable to reconstruct the details in
his chains of reasoning Laplace would say, “It is
easy to see that…” and give the result with out
any further explanation.
– One astronomer observed, “I never came across one
of Laplace’s ‘Thus it plainly appears,’ without feeling
sure that I had hours of hard work before me to fill up
the chasm and find out and show how it plainly
appears.”
Bernouilli
• Lucky Luke walks into Cæsar’s Palace in
Vegas and sees a new game called only
“The Coin.” At this game, Luke is given a
coin to toss. As long as tails show, the
game continues; once heads shows the
game is over and Luke collects his
winnings. If heads shows on the 1st toss,
Luke receives \$1, on the 2nd toss he
receives \$2, 3rd \$4, 4th \$8, and so on.
Formula for Expectation
• E = a1p1 + a2p2 + … + anpn
Bernouilli
• Lucky Luke walks into Cæsar’s Palace in
Vegas and sees a new game called only
“The Coin.” At this game, Luke is given a
coin to toss. As long as tails show, the
game continues; once heads shows the
game is over and Luke collects his
winnings. If heads shows on the 1st toss,
Luke receives \$1, on the 2nd toss he
receives \$2, 3rd \$4, 4th \$8, and so on.
• Luke is not called “Lucky” for nothing, but
he isn’t sure he has enough money to play
the game. Assuming this is a fair game
(i.e. cost to play = expected winnings),
how much does the game cost to play?
Laplace
• Wrote a book to aquaint a broader circle
of readers with the fundamentals of
prbability theory and its applications
without resorting to higher mathematics.
• Gives definition of the probability of an
event as:
– Pr[event] = # of favorable outcomes / total #
of outcomes
Mutually Exclusive and
Independent Events
• 1) If A and B are mutually exclusive events
(they cannot both happen at the same
time) then,
– Pr [A or B] = Pr[A] + Pr[B]
• 2) If A and B are two independent events
(the occurance of one doesn’t affect the
probability of the other) then,
– Pr[A and B] = Pr[A] Pr[B]
Bernoulli’s Formula
• If the probability of successes on a single
trial is denoted by p and the probability of
failure by q = 1-p, then p and q remain
constant from trial to trial.
• Bernoulli showed that the probability of
observing exactly r successes in n trials
was expressed by the rth term of the
expansion for (p+ q)r:
– Pr[r successes and n-r failures] = (nCr) pr qn-r

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