Footnote to my impending post about the history of value and utility:
After Pascal’s and Fermat’s work on the problem of points, and Huygens’s work on expected value, the next major work on probability was Jakob Bernoulli’s Ars conjectandi, written between 1684 and 1689 and published posthumously by his nephew Nicolaus Bernoulli in 1713. Ars conjectandi had 3 important contributions to probability theory:
[1] The concept that expected experience is conserved, or that probabilities must sum to 1.
Bernoulli generalized Huygens’s principle of expected value in a random event as
E=p1∗a1+p2∗a2+p3∗a3p1+p2+p3
[ where pi is the probability of the ith outcome, and ai is the payout from the ith outcome ]
and said that, in every case, the denominator—i.e. the probabilities of all possible events—must sum to 1, because only one thing can happen to you
[ making the expected value formula just
E=p1∗a1+p2∗a2+p3∗a3
with normalized probabilities! ]
[2] The explicit application of strategies starting with the binomial theorem [ known to ancient mathematicians as the triangle pattern studied by Pascal
and first successfully analyzed algebraically by Newton ] to combinatorics in random games [which could be biased] - resulting in e.g. [ the formula for the number of ways to choose k items of equivalent type, from a lineup of n [unique-identity] items ] [useful for calculating the expected distribution of outcomes in many-turn fair random games, or random games where all more-probable outcomes are modeled as being exactly twice, three times, etc. as probable as some other outcome],
written as (nk):
(nk)=n!k!(n−k)!
[ A series of random events [a “stochastic process”] can be viewed as a zig-zaggy path moving down the triangle, with the tiers as events, [whether we just moved LEFT or RIGHT] as the discrete outcome of an event, and the numbers as the relative probability density of our current score, or count of preferred events.
When we calculate (nk), we’re calculating one of those relative probability densities. We’re thinking of n as our total long-run number of events, and k as our target score, or count of preferred events.
We calculate (nk) by first “weighting in” all possible orderings of n, by taking (n∗(n−1)...), and then by “factoring out” all possible orderings k! of k ways to achieve our chosen W condition [since we always take the same count k of W-type outcomes as interchangeable], and “factoring out” all possible orderings (n−k)! of our chosen L condition [since we’re indifferent between those too].
[My explanation here has no particular relation to how Bernoulli reasoned through this.] ]
Bernoulli did not stop with (nk) and discrete probability analysis, however; he went on to analyze probabilities [in games with discrete outcomes] as real-valued, resulting in the Bernoulli probability distribution.
[3] The empirical “Law of Large Numbers”, which says that, after you repeat a random game for many turns and add up all the outcomes, the total final outcome will approach the number of turns, times the expected distribution of outcomes in a single turn. E.g. if a die is biased to roll
a 6 40% of the time
a 5 25% of the time
a 4 20% of the time
a 3 8% of the time
a 2 4% of the time, and
a 1 3% of the time
then after 1,000 rolls, your counts should be “close” to
and even “closer” to these ideal ratios after 1,000,000 rolls
- which Bernoulli brought up in the fourth and final section of the book, in the context of analyzing sociological data and policymaking.
One source: “Do Dice Play God?” by Ian Stewart
[ Please DM me if you would like the author of this post to explain this stuff better. I don’t have much idea how clear I am being to a LessWrong audience! ]
Footnote to my impending post about the history of value and utility:
After Pascal’s and Fermat’s work on the problem of points, and Huygens’s work on expected value, the next major work on probability was Jakob Bernoulli’s Ars conjectandi, written between 1684 and 1689 and published posthumously by his nephew Nicolaus Bernoulli in 1713. Ars conjectandi had 3 important contributions to probability theory:
[1] The concept that expected experience is conserved, or that probabilities must sum to 1.
Bernoulli generalized Huygens’s principle of expected value in a random event as
E=p1∗a1+p2∗a2+p3∗a3p1+p2+p3
[ where pi is the probability of the ith outcome, and ai is the payout from the ith outcome ]
and said that, in every case, the denominator—i.e. the probabilities of all possible events—must sum to 1, because only one thing can happen to you
[ making the expected value formula just
E=p1∗a1+p2∗a2+p3∗a3
with normalized probabilities! ]
[2] The explicit application of strategies starting with the binomial theorem [ known to ancient mathematicians as the triangle pattern studied by Pascal
and first successfully analyzed algebraically by Newton ] to combinatorics in random games [which could be biased] - resulting in e.g. [ the formula for the number of ways to choose k items of equivalent type, from a lineup of n [unique-identity] items ] [useful for calculating the expected distribution of outcomes in many-turn fair random games, or random games where all more-probable outcomes are modeled as being exactly twice, three times, etc. as probable as some other outcome],
written as (nk):
(nk)=n!k!(n−k)!
[ A series of random events [a “stochastic process”] can be viewed as a zig-zaggy path moving down the triangle, with the tiers as events, [whether we just moved LEFT or RIGHT] as the discrete outcome of an event, and the numbers as the relative probability density of our current score, or count of preferred events.
When we calculate (nk), we’re calculating one of those relative probability densities. We’re thinking of n as our total long-run number of events, and k as our target score, or count of preferred events.
We calculate (nk) by first “weighting in” all possible orderings of n, by taking (n∗(n−1)...), and then by “factoring out” all possible orderings k! of k ways to achieve our chosen W condition [since we always take the same count k of W-type outcomes as interchangeable], and “factoring out” all possible orderings (n−k)! of our chosen L condition [since we’re indifferent between those too].
[My explanation here has no particular relation to how Bernoulli reasoned through this.] ]
Bernoulli did not stop with (nk) and discrete probability analysis, however; he went on to analyze probabilities [in games with discrete outcomes] as real-valued, resulting in the Bernoulli probability distribution.
[3] The empirical “Law of Large Numbers”, which says that, after you repeat a random game for many turns and add up all the outcomes, the total final outcome will approach the number of turns, times the expected distribution of outcomes in a single turn. E.g. if a die is biased to roll
a 6 40% of the time
a 5 25% of the time
a 4 20% of the time
a 3 8% of the time
a 2 4% of the time, and
a 1 3% of the time
then after 1,000 rolls, your counts should be “close” to
6: .4*1,000 = 400
5: .25*1,000 = 250
4: .2*1,000 = 200
3: .08*1,000 = 80
2: .04*1,000 = 40
1: .03*1,000 = 30
and even “closer” to these ideal ratios after 1,000,000 rolls
- which Bernoulli brought up in the fourth and final section of the book, in the context of analyzing sociological data and policymaking.
One source: “Do Dice Play God?” by Ian Stewart
[ Please DM me if you would like the author of this post to explain this stuff better. I don’t have much idea how clear I am being to a LessWrong audience! ]