Week 1: Classical Probability (17-18 C)

Question 1

Roots of Classical probability?

Answer 1

General concepts had existed as aleatory contracts since antiquity. (insurance, gambling, annuities)

• First formulations occur around 17th century
o 1654: Blaise Pascal & Pierre Fermat begin discourse on subject
o Note on context: At the time, the idea of a “mathematician” did not exist, so both were natural scientists, and made contributions to other areas such as theology
o Fermat writes to Pascal, concerning the division of stakes in gambling
 Context: Gambling was aristocratic entertainment, not a sin

Question 2

What did Pascal/Fermat discuss? When is this?

Answer 2

Gambling - 17th century

Suppose two players (‘A’ and ‘B’) are playing a game of dice. Rolls of 1-3 count as a point for A, while rolls of 4-6 give a point to B. Each player puts in $32, and after four rolls either the player with the most points wins $64, or in the case of a tie each player gets back $32.

• Fermat’s Question: Suppose after three rolls, A leads 2:1, and for some reason the game ends without completing the fourth roll. How should the stakes be divided?

Pascal answers - equiprobable either wins next toss, so winnings should be half of A wins all and half of tie
>today’s “expected value”

Question 3

Pascal’s Theological Wager

Answer 3

Gain of heaven infinite, loss of hell infinite. Believe in God since E(x) is so skewed

Question 4

How was probability used in the 17th, 18th centuries?

Answer 4

A tool for decision making - probabilities never existed in isolation; always with its expected outcome.

Typical analysis was repeated chance events (e.g. Bernoulli trials & Math of Fair Games)

Question 5

Describe the work of Jacob Bernoulli.

Answer 5

Analyzed “Bernoulli Trials” during 18th C in Switzerland
o performed analysis based on many repeated coin flips
o Worked to solve problems such as the probability of getting “h” heads in “n” tosses

Question 6

Shortcomings of Bernoulli’s work? How was it resolved?

Answer 6

o Chance events are rarely equiprobable in nature
o We can rarely know the probability in advance

Many, e.g. Abraham de Moivre suggested we should use empirical data to predict, since we can’t know a priori the probabilities
o 1725: Publishes “Of the Valuation of Annuities for Life”

• Thomas Bayes and Simon Laplace also worked to make probability practically usable
o Tried to estimate real probabilities from finite realizations
o Instead of direct probability a la Bernoulli, they worked with inverse probability, using the results to infer the underlying probabilities

 Using empirical data, we can improve our estimates of the causal probabilities
• Provides a statistical inference

Question 7

Pascal?

Answer 7

• answers Fermat’s gambling question with early version of E(x)
• Pascal’s Theological wager

Question 8

Fermat?

Answer 8

-begins discourse with Pascal on gambling

Question 9

Abraham de Moivre?

Answer 9

o 1725: Publishes “Of the Valuation of Annuities for Life”
 Establishes that we cannot predict life in advance. Instead, we should rely on empirical data of the past to estimate future behaviour
• Idea is adopted through 18th century

Question 10

Thomas Bayes

Answer 10

Along with Pierre-Simon Laplace, works with conditional and inverse probability to use empirical data to improve causal probability estimates

Question 11

Pierre-Simon Laplace

Answer 11

Along with Thomas Bayes, works with conditional and inverse probability to use empirical data to improve causal probability estimates

Question 12

Ontology?

Answer 12

The study of knowledge; what we believe to be true

Question 13

Epistemology?

Answer 13

The theory of human understanding; what human knowledge is capable of