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The Math Learning Portfolio Casino: Test Your Luck

Probability Laws
Four Probability Properties
Pascal's Wager and Bernoulli Trials
John Forbes Nash and Game Theory
Chance Games
Math in Music
Demonstration of Goals

Pascal's Wager and Bernoulli Trials

Blaise Pascal and Jakob Bernoulli

Blaise Pascal (1623-1662)

Pascal's Wager

Pascal’s Wager is an argument that proves God’s existence, according to Blaise.


"Let us examine this point and declare: 'Either God exists, or He does not.' To view shall we incline? Reason cannot decide for us one way or the other: we are separated by an infinite gulf. At the extremity of this infinite distance a game is in progress, where either heads or tails may turn up. What will you wager? According to reason you cannot bet either way; according to reason you can defend neither proposition . 'Both are wrong. The right thing is not to wager at all.' Yes, but a bet must be laid. There is no option: you have joined the game."


                                                                                               - Blaise Pascal


In mathematical terms, Pascal’s Wager can be expressed in the following expression:

    e x 2/e – (1 – e) x 1 = 1 + e


Let e be the probability of God’s existence (This number cannot be zero; for example it can be an extremely small number). Let 1 be the measure of happiness.


The expression for not believing in God, whether he exists or not is:

    (1 – e) x 1 + e x 0 = 1 – e


Since 1 + e > 1 – e, so to maximize happiness, one should choose to believe God.





            In today’s society, there are many individuals who try to persuade others into believing their morals and ideas. There can be truth to what they say, as well, many people add their own thoughts to create a revolutionary product. Pascal’s Wager was part of fragments of a manuscript, and is not complete. His main ideas were about the Bible and Christianity, and he added his own thoughts based on philosophy. His thoughts may have been influenced by his family status, as at the time his mother had passed away, and his father was sick. Therefore, there is no proof that his arguments are factually correct. Just like modern authors, writers tend to make non-fiction become fiction.

However, now these types of books can be considered science fiction, fantasy, or fiction. There are also books that contain historical and religious information, and the author adds his or her ideas to form a plotline. An example is The Da Vinci Code, by Dan Brown. Brown uses historical information of the Opus Dei and the Priory of Sion. Then he sets his characters up to reveal the secret behind the Holy Grail.

In addition, a closer to relation to Pascal’s Wager would be writing collections from philosophers. Morals and historical or religious facts are used as the basis of many discussions.





      Blaise Pascal was born in Clermont, France, on June 19, 1623. His father, Etienne Pascal did not follow traditional educational views, and therefore had Blaise home schooled. Blaise’s father believed that Pascal should not study mathematics before age 15, and removed all mathematics related books from the house. At the young age of 12, Pascal taught himself geometry and discovered that the sum of the interior angles of a triangle summed up to be 180 degrees. At age 16, Pascal created his own geometry theorem, called Pascal’s mystic hexagon. Blaise Pascal was a famous French mathematician, philosopher, religious writer, as well as a scientist.


Early Inventions


In late 1639, the Pascals left Paris and moved to Rouen where Etienne worked as a tax collector. Soon, Pascal wrote the Essay on Conic Sections and had it published in 1640. Furthermore, Pascal invented a digital calculator to ease his father’s work. He named it the Pascaline. Soon however, problems aroused due to the stature of the French currency. Around 50 prototypes were created, but few calculators were sold.

            Pascal’s father wounded his leg in 1646, and Pascal soon became religious. In 1647, Pascal worked with experiments about atmospheric pressure. He proved that a vacuum existed. Descartes, a scientist, disagreed with him, and said Pascal “… [Had] too much vacuum in his head”. In September 1651, Pascal’s father passed away, and Pascal’s Christian beliefs to death inspired him to write his book Pensées. Pascal also worked on the Treatise on the Arithmetical Triangle, the Pascal Triangle, conic sections, the cycloid, binomial coefficients, as well as theorems in projective geometry.

     Pascal’s writings in Pensées are composed of over one thousand sections regarding philosophic and religious matters. Since his father’s injury, Pascal became a Christian, and he preaches his opinions in this collaboration of his works. Penser in French means ‘to think’.

            In cooperation with Fermat, another mathematician, Pascal created the theory of probability. Pascal’s famous philosophic creation, Pensées, contains Pascal’s Wager, a claim to prove God’s existence. Pascal argued, “If God does not exist, one will lose nothing by believing in him, while if he does not exist, one will lose everything by not believing”.

The Pascaline Digital Calculator

Jakob Bernoulli (1654-1705)

Probability Equation


            Jakob (Jacques or James) Bernoulli was born on December 27, 1654, in Basel, Switzerland, in a Protestant family. His father, Nicolaus Bernoulli ran a spice business, inherited from his own father. The Bernoullis fled from Spanish rulers at the time. Nicolaus was a magistrate, and important member of the town. Jakob’s mother was from a family of bankers.

Bernoulli studied theology due to his father’s will, but later studied mathematics and astronomy. He received a degree in theology in 1676, and prior to that, a master’s degree in philosophy in 1671. Jakob’s brother, Johann Bernoulli, also studied mathematics, and the two often competed to see who could solve more difficult problems faster. The two brothers are the most significant founders of calculus, other than Newton.

            Bernoulli traveled to England, France, and Holland to broaden his knowledge of comets and gravity. Later, he became a professor of mathematics at the University of Basel. In 1683, he taught the mechanics of solids and liquids.




Although Bernoulli and his brother, Johann, often disputed over who was superior in discovery and mathematics, both worked diligently and published works on probability and geometry, from 1685 to 1687.

Jakob came up with properties of the Bernoulli Numbers, and wrote the Ars conjectandi. This was the first treatise about probability, which included the law of large numbers, theories of permutation and combination, binominal theorem (for positive and negative integers), multinomial theorem, and the Bernoulli differential equation. Bernoulli also solved the brachistochrone problem.

In 1689, Bernoulli published a book on infinity, and later discovered a rule to determine evolutes of a curve. Later on in his life, he studied caustic curves, parabolas, logarithmic spirals and epicycloids.


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