First Notes on Probability


A footballer kicks the ball hard and straight towards its target, the goal. The goalie has seen it and is ready in place to block and collect it. But it accidentally hits a defending player and ricochets into an unprotected corner of the goal…is that “Chance”? Or “Luck”? Good luck for one side, bad luck for the other…? (a real case from ISL 2021 but common enough everywhere…)

How fragile is life…a teenager cycles to her Sanskrit exam…is killed by a lorry trying to escape corrupt police… (a real case from 2017)

[Footnote Is it (objectively) worse that the teenager killed was cycling to a Sanskrit exam and not, say, a drug dealer? It is… Is it worse that the lorry killed her not due to some unpredictable mechanical failure but because it was avoiding corrupt police? It is..
A lorry kills a drug dealer accidentally due to mechanical failure isn’t as bad as if due to corruption it kills a girl going to an exam; a lorry accidentally kills a drug dealer is different from the same lorry due to corruption killing a girl going to her Sanskrit exam… What happened @WBPolice @MamataOfficial to the lorry driver who killed the girl cycling to her Sanskrit exam?…
Sanskrit a hard rich esoteric subject, the girl cycling to her Sanskrit exam must have been the pride of her family… The family expected her to come home safely, talk of how hard it was, aim for college, become a teacher, a mother herself… Instead she was killed by a speeding lorry trying to escape paying a bribe to the police…the family in sudden shock…]

The girl cycling to her Sanskrit exam, killed by a lorry trying to escape corrupt police. Is that “Chance” or “Luck”? Luck is Chance without any probability distribution

…Luck is Chance without any probability distribution…, I say… (and am reminded of Milton Friedman telling me decades ago he believed such a thing existed as luck, good or bad… )… It is bad luck for one team, good luck for the other that the ball ricochets off the defensive player into the goal… There is no probability distribution… Or is there one lurking beneath? Can a team through, say, much practice, reduce its “bad luck” in defensive situations? Similarly, the girl killed cycling to her Sanskrit exam by a speeding lorry escaping corrupt police, is it a “one-off” event of bad luck or is there a probability distribution lurking beneath? Viz., if governance improved, corruption was reduced, the chance of such deaths falls?

These are two disparate cases of what might be called Random Events…. where there is no underlying probability distribution at least prima facie… (contrast the “Random Variable” usually seen in statistical/stochastic theory!)…

a Random Event is one of Chance without any probability distribution discernible prima facie…

if upon inspection/scrutiny/study we think we find a probability distribution, it’s no longer a case of Chance, Randomness, Luck…but one of Causation…

There’s a world of difference between saying that B was a random event & saying that B was the result of or caused by something else called A…

I.e. there’s Randomness/Chance/Luck in one corner without any probability distribution possible to be discerned; and there’s Causation and Probability in the opposite corner…

I.e., a Random Event/case of Chance/Luck is one of 0 ex ante or prima facie probability… No one expected it or *could have* expected it => inspection/scrutiny/study follows:

— if that yields nothing, it remains Random

— if a probability distribution is detected/discerned, there’s Causation…

The end result of an inspection/ scrutiny/study is then

Either “We found nothing; B is a genuine Random Event”

Or “We found A might have caused B” …

“Chance visits the prepared mind” said Pasteur famously… if you can work to reduce your “bad luck”, can you also work to improve your “good luck”? Perhaps sometimes in each case… Perhaps where Statistical theory went wrong ab initio, is starting with a “Random Variable”… I start with a Random Event instead…

start with a Random Event, not the Random Variable of usual stochastic/statistical theory…

A Random Event is one which no-one in the population has placed a non-zero ex ante/prior probability on, ie everyone has placed a 0 ex ante/prior probability on… this includes finding the possibility unimaginable. Eg, no one could have imagined the girl cycling to her Sanskrit exam would be killed by a lorry fleeing corrupt police.

Tackling probability theory seriously, using Keynes Jeffreys IJ Good Emile Borel and everyone else, is going to be very hard … stretching from good or bad luck in human situations to positions of particles in Quantum Mechanics… maybe a hundred different cases like the two above will illustrate…

2017 Chapter 3 Facts, Hypotheses, and Guesses

3.1.  A scientific fact is something for which the evidence is overwhelming, to deny which would be naive or foolish or unreasonable.

3.2.  Two central facts of physics established over the centuries are the heliocentric solar system & the spherical Earth (its curved surface).

3.3.  Other scientific facts we certainly know today include that we inhale oxygen needed for our metabolism and exhale carbon dioxide as a waste, that oxygen enters & carbon dioxide exits through the lungs, that both are transported everywhere in our mammalian bodies by the blood pumped by the  heart, and so on.

3.4. It is best to keep quite small the number of such illustrations of certain scientific knowledge for reason of Rutherford’s Common Man: “An alleged scientific discovery has no merit unless it can be explained to a barmaid”.  What the barmaid and Rutherford share is rationality, reasonableness, comprehension, language.  Perhaps pubs in college towns should supply bartenders to witness doctoral viva voces in science and everything else…

3.5. Chapter 6 of my 1989  Philosophy of Economics is titled “Expertise & Democracy”.  It doesn’t say the Common Man should judge quality but that the road for the Common Man to do so should be open.  The road to judge whether an advance has been made, a contribution to knowledge, new facts established, should be open to everyone including Rutherford’s barmaid.

3.6. Many people have not thought about the flatness or curvature of Earth’s surface, nor know about their own respiration & metabolism.  (The easiest way to improve a nation’s health incidentally, may be by greater general understanding of the science of respiration, metabolism  etc.) Many people especially in India & China endorse a geocentric solar system today when they act as they do believing in astrology’s horoscopes. (Astrology could be only a trivial pastime, or at least we have to hope so as believers may include those purporting to science!)  Russell cynically said “Many people would sooner die than think, in fact they do”.

3.7.  Other facts can be deduced or conjectured from central facts like heliocentrism.  Eddington noted we are transported by the Earth & Sun: the Earth as our vehicle we travel at 20 miles per second around the Sun which in turn carries us at 12 miles a second through the galactic system, which carries us at 250 miles a second amid the spiral nebulae…

3.8.  Eddington’s 1927 Gifford lectures *The Nature of Physical Reality* had a laudable if unsuccessful aim of expressing in words Electromagnetism, Relativity, Entropy, Gravitation, Quantum Theory, all the new physics!  A member of the audience asked about relativity “You must find the journey between Cambridge & Edinburgh tiring.  I can understand the fatigue if you travel to Edinburgh but why are you tired if Edinburgh travels to you?”.  Physics isn’t without its humour.  Eddington had to reply, “If motion could tire, we ought to be dead tired” referring to our carriage by the Earth, Sun, galaxies.

3.9.  Beyond relatively few central facts, physics has uncountable multitudes of “surmised facts” consisting of hypotheses & conjectures,  guesses & speculations.

3.10.  Boundaries between facts and hypotheses or between hypotheses and guesses need not be hard and fast or dogmatically precise…  We can always ask in a given case, is this a known accepted fact or a surmise or just an informed guess, what is the situation?  Earth’s curved surface or the heliocentric solar system are established facts; Eddington’s calculations of  speeds at which we travel thru galaxies are less so.

3.11.  Between hypotheses and guesses the boundaries may be imprecise again but with a broad difference: With guesses there is no known empirical or even conceptual test available or possible, with hypotheses some test giving a yes/no answer may be possible.

3.12  At least 1/3 of all stars is a “double” or two stars together with no planets, reports Eddington from the important research of his colleague JH Jeans.  Jeans concluded double star formation to be typical,  formation of a solar system to be “a freak”.   Jeans’s theory of our solar system’s formation is that a thousand million years ago another star passed not far from what became the planet Neptune, this star passing by caused “big protruberances” of the Sun, big enough for the formation of the planets by eruption from the Sun.

3.13.  JH Jeans’s description of how another star once passed by our Sun causing it to disgorge the planets of the solar system is a guess or speculation which can’t be empirically tested.

3.14.   Yet Jeans’s guess is reasonable & comprehensible, following as it does a line of reasoning from less uncertain facts. As such it is more plausible than if we instead guessed the planets emerged because the Sun sneezed like a human.

3.15.  Jeans and Eddington concluded the chances of planets arising from any star are remote.  Compound that with the chances of water, photosynthesis, oxygen, life arising,  and the likelihood of a Da Vinci elsewhere in the Universe becomes remoter still.  Humankind may be alone (is my surmise)!

3.16.  A contemporary view from Helen Klus is “There may be millions of other planets in our galaxy that contain life”.  If so Jeans and Eddington were wrong, and I too to believe them.

3.17.  Popper proposed we try to falsify our guesses and findings to come to know what is scientific truth at a given time, that we adopt a critical attitude to our own and others’ scientific claims.  Everything we think we know we seek to re-examine critically if we are to progress with and advance our knowledge.

3.18.  Jeans conjectures a star passed by causing the Sun to disgorge planets a 1000 million years ago?  Popper says look at the reasoning and facts again.

3.19.   A youngster is sceptical seeing a ship disappear over the horizon.  Good.   Bring him/her back the next day & the next, to see more ships doing the same… But how do we know the ships aren’t just falling off the edge? the child asks… Excellent question… Let’s take a helicopter, chase after it.  Or imagine we do so notionally if we can’t afford one.  “We haven’t got the money, so we’ll have to think” (Rutherford).  Ok, says the child, I accept the Earth’s surface seems curved in this region, but how can we tell it is so everywhere? To Jules Verne!

3.20.  The vast bulk of physical facts may be surmises midway between guesses/speculations on one hand, and known central facts on the other. This bulk of surmised facts I describe as hypotheses/conjectures open to hypothesis testing: offering yes/no results to empirical or even just conceptual experiments…

3.21.  Heisenberg’s  “thought experiment” led to his declaring we can know the location or velocity of a particle as precisely as we want but not both together at the same time… Heisenberg’s superb *Principles* is the exemplary doctoral thesis of the star pupil of Sommerfeld… You describe one two three four extant empirical experiments that you say everyone finds puzzling, inadequately explained, anomalous…You provide your criticism of the existing theory or theories used in these experiments, then propose a theory to replace them…Then you discuss possible criticisms and extensions of the theory you’ve newly offered. Then apply it to reexamining the original empirical experiments.  Finally you place in an appendix the “Mathematical Apparatus” you’ve used, mathematics in its place as the servant not master of physics.

3.22.  Heisenberg admits with utmost honesty after all his new theory that most of the empirical experiments remain explicable classically!

[GEMoore:  “You aren’t denying the electron exists, you’re saying we can’t tell where it is.”

Heisenberg: “Exactly so! (Well, almost!)”.]

3.23   Eddington suggested a “thought experiment” of an empty vessel with a sliding door halfway: in the left half is air; the closed right half is a vacuum; opening the sliding door causes the air molecules to diffuse thru the vessel, hitting each other & the walls randomly, never again congregating back to the initial left side.  Eddington used it to show increasing entropy, “the running down of the Universe”, a “loss” in the initial “organization” of the molecules..

3.24.  Complicated experiments on a hard subject were those of Clauser, and later Aspect, seeking to test Bell’s theorem in the contest between Einstein and the Quantum Mechanics of the “Copenhagen School”.   The latter “won”, for the time being at least.  But Popper’s scientific method would require them and everyone else to think again!  Try to refute your own findings!  Was Bell’s theory adequate?  Were the Clauser and Aspect experiments adequate?

3.25.   John Wisdom gave a deliberately easy empirical experiment with a yes/no answer: a room seems empty, nothing to see feel hear smell taste in it.  A declares “It’s empty”, B declares “It’s not”, throws a lighted match in it, there’s an explosion. “Gas” says B….

3.26.   Empirical questions should be nice enough to provide us a binary choice of answers: Yes/No; True/False; or, if we like, probability ≈ 1 probability ≈ 0.  Until a binary choice is available the question is not purely empirical but still theoretical.  John Wisdom was keen to argue that even *after* all facts, all empirical results, are in, there may still not be a conclusion until we have  *reflected* about all that’s before us.

3.27  Other than probability ≈ 1, probability ≈ 0, the only probability number   of interest is the halfway point 0.5, which is, as it were a point of “Ignorance” or “Indifference”, an *unstable* point which must move either towards 1 or towards 0, ie towards certain knowledge in one direction or other.

3.28.  I suggest physics define

probability ≈ 1 as our *rational degree of belief* in the heliocentric solar system,

probability ≈ 0 as its contrary….

Or we can normalize with probability ≈ 0 as the Earth being flat, probability ≈ 1 its contrary.

3.29.  JH Jeans’ surmised description of the formation of the solar system (of how another star once passed by our Sun causing it to disgorge the planets) is a guess we may place at probability = 0.5.

3.30.  My proposal is, in other words, that we calibrate our rational degrees of belief by certainty eg heliocentrism prob ≈ 1, by JH Jeans’s Guess prob =0.5, and a lower limit of certainty again prob ≈ 0 eg the Earth being flat.

3.31. Certain knowledge  prob ≈ 1 is not “absolute knowledge” prob = 1.  Our slogan remains “This is where we got to”.  It is not *logically* impossible Copernicus & everyone got it wrong… Before Copernicus, Ptolemy was certain knowledge.  What can be certain knowledge in one era may be expected to change as evidence advances with time.

3.32.  Calibrating with certainty prob ≈ 1, a Guess prob=0.5, a lower limit of certainty again prob≈0, every other case must be compared and contrasted to these three in turn.  Moving  from a Guess at probability 0.5 positively towards certain knowledge prob ≈ 1 or negatively towards certain knowledge prob ≈ 0 is the process of hypothesis testing given improving evidence.  Our confidence in our belief increases away from 0.5 towards 1, or decreases away from 0.5 towards 0… call it positive, negative confidence respectively.

3.33.  A person applies for a job, he’s at 0.5, doesn’t know if he will get it…As evidence improves his confidence moves  positively towards 1 or negatively towards 0.  The initial Ignorance point 0.5 is unstable: it’s as if the 0.5 point belongs to Leibniz and movement in directions positive to 1 or negative to 0 both belong to Bayes.

3.34.  We may have the rudiments of the probability theory that eluded Keynes Ramsey de Finetti  Good etc…who may have at different stages in different ways confused objective and subjective.  (In turn, Quantum Mechanics theorists, not having time to investigate probability themselves, may have confused themselves with the dogmas of probability theorists!)

3.35.  Ramsey said we bet all the time in life: we go to the railway station betting the train will depart…or we would have stayed home… In my terms we go to the railway station betting the train will depart because we place the probability above our JH Jeans Guess of 0.5, below the probability of heliocentrism ≈ 1

3.36.  Two commuters place a probability 0.9 each day of the train departing; one day A doesn’t turn up, B does & finds the train doesn’t run. B asks A the next day, she says yes my son told me the train wasn’t going to run because there’d been a mishap down the line so I stayed home.  A’s assessment of the evidence had changed due to the new information hence her rational degree of belief, her probability, changed too… In Bayesian terms, A’s “prior probability” 0.9 the train would run crashed to a “posterior probability” ≈ 0 based on her son’s information.

3.37  From the Ignorance point 0.5 we act in one direction as prob -> 1, we act in the opposite direction as prob -> 0.   Our confidence in our belief increases away from 0.5 towards 1, or away from 0.5 towards 0… positive or negative confidence.

Alternatively, the 0.5 point is a point of Ignorance, 1 and 0 are the points of Rational Belief and Rational Disbelief respectively….

Alternatively, probability  1 is  T in “truth tables”, probability 0 is F…. 0.5 is, say, I for Ignorance or Indifference, an unstable point, moving with learning towards either T or F.

Rational Belief______Ignorance_______Rational Disbelief
Probability ≈  1                =  0.5                        ≈ 0
True T                           Ignorance I                False F

We rationally believe in heliocentrism, rationally disbelieve the Earth is flat, & we guess JH Jeans’ theory of planet formation is correct.  Once we have three such points, every other case of belief/disbelief that arises for anyone may be compared & contrasted with each of these

3.38  The 0.5 Guess point is the unstable one of “insufficient evidence”; moving in either direction increases evidence with hypothesis testing.  Moving  from a Guess at prob 0.5 positively towards certain knowledge prob ≈ 1 or negatively towards certain knowledge prob ≈ 0 is the process of hypothesis testing given improving evidence.   “It is wrong in all cases to believe on insufficient evidence…” said Clifford.  (Clifford, dead at 33 in 1879, the same year Einstein is born, anticipated Einstein on gravitation by a generation, and Einstein knew it in his Berne years.)

3.39  Games of chance like rolling dice or choosing a card from a shuffled pack are artificial human contrivances to simulate states of ignorance.

In my terms, a 6-sided fair die before it is rolled would be represented by a six element vector (0.5, 0.5, 0.5, 0.5, 0.5, 0.5); its possible outcomes are the rows of a 6×6 identity or unit matrix (1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1).

With 52 cards, obviously if we know the initial order & how precisely it is shuffled (say in slow motion by a machine) we know the new order.  Our ignorance is deliberately contrived in a pack of cards to create randomness & chance for amusement; it may be a mistake to think nature is like that.

In a randomly shuffled pack, what is the top card is in my terms a 52 dimensional vector of (0.5) s, 52 possible results by a 52 dimensional identity or unit matrix.

The 0.5 would still represent the halfway between 1 and 0 in each case but otherwise = 1/6 is an ex ante “probability” in case of the die, 1/52 in case of the cards.

3.40  Einstein saying (if he did) of Quantum Mechanics “God does not roll dice” may have just meant to say: “Don’t confuse our contrived games of chance for our real ignorance of Nature…!”.   Max Born, Heisenberg and others may have tossed in prevalent probability ideas/dogmas without having the time to think them all through…