Probability by 2 Great Friends

Today Probability is a “money” Math, used in Actuarial Science, Derivatives (Options) in Black-Scholes Formula.

In the beginning it was “A Priori” Probability by Pascal (1623-1662), then Fermat (1601-1665) invented today’s “A Posteriori” Probability.

“A Priori” assumes every thing is naturally “like that”: eg. Each coin has 1/2 chance for head, 1/2 for tail. Each dice has 1/6 equal chance for each face (1-6).

“A Posteriori” by Fermat, then later the exile Protestant French mathematician De Moivre (who discovered Normal Distribution), is based on observation of “already happened” statistic data.

Cardano (1501-1576) born 150 years earlier than Pascal and Fermat, himself a weird genius in Medicine, Math and an addictive gambler, found the rule of + and x for chances (he did not know the name ‘Probability’ then ):

Addition + Rule: throw a dice, chance to get a “1 and 2” faces:
1/6 +1/6 = 2/6 = 1/3
(Correct: 1 & 2 out of other six faces)

Multiplication x Rule: throw two dices, chance to get a “1 followed by a 2 ” faces : 1/6 x 1/6 = 1/36
(Assume 1st throw does not affect or influence the 2nd throw: independent events)

Pascal never met Fermat personally, only through correspondences (like emails today), but Pascal regarded with respect Fermat as superior in Math than himself.

Fermat and Descartes were not so. Descartes openly criticized Fermat as a second class mathematician. Both independently discovered Analytical Geometry, but Descartes scored the credit in ‘Cartesian’ coordinates.

Fermat never published any books in Math. As a successful judge in Toulous, he spent his free time as an amateur mathematician, especially in Number Theory. He showed his Math discoveries to friends in letters but never provided proofs. Hence the Fermat’s Last Theorem made the world mathematicians after him (Gauss, Euler, Kummer, Sophie Germaine, Andrew Wiles…) busy for 380 years until 1994.

Fermat died 3 years after Pascal. Another Modern Mathematics was being born – Calculus – in UK by Newton and Germany by Leibniz. Probability was put in the back seat over-taken by Calculus.

Logic: Pascal Wager

Pascal Wager:

1. We can choose to believe God exists, or we can choose not to so believe.
2. If we reject God and act accordingly, we risk everlasting agony and torment if He does exist (Type I error in Statistics lingo) but enjoy fleeting earthly delights if He doesn’t exist.
3. If we accept God and act accordingly, we risk little if He doesn’t exist (Type II error) but enjoy endless heavenly bliss if He does exist.
4. It’s in our self-interest to accept God’s existence.
5. Therefore God exists!

Mathematical Proof:
Pascal assumed
Probability of God exists = p
Probability God doesn’t exist = 1-p

You lead 2 lives, either Worldly (世俗) or Piously (虔,诚) , you get rewards X, Y, infinity or Z, as shown in table below.
In Worldly Life, the Expectation in probability is
Ew = p.X + (1-p).Y
In pious life, the Expectation is
Ep = p.∞  + (1-p).Z
Regardless how big the rewards in X (negative, punishment by God) ,Y, Z (self-sacrifice, negative reward)
Ep is infinitely bigger than Ew.
(Ew is good only if p= 0)
Conclusion: no matter what, pious life is better.
pascal