Week 1: Classical Probability (17-18 C) Flashcards Preview

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Flashcards in Week 1: Classical Probability (17-18 C) Deck (13)
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1
Q

Roots of Classical probability?

A

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

2
Q

What did Pascal/Fermat discuss? When is this?

A

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”

3
Q

Pascal’s Theological Wager

A

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

4
Q

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

A

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)

5
Q

Describe the work of Jacob Bernoulli.

A

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

6
Q

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

A

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

7
Q

Pascal?

A
  • answers Fermat’s gambling question with early version of E(x)
  • Pascal’s Theological wager
8
Q

Fermat?

A

-begins discourse with Pascal on gambling

9
Q

Abraham de Moivre?

A

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

10
Q

Thomas Bayes

A

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

11
Q

Pierre-Simon Laplace

A

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

12
Q

Ontology?

A

The study of knowledge; what we believe to be true

13
Q

Epistemology?

A

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