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{{a| | {{a|systems|{{image|Quincunx|jpg|Some independent events, yesterday}}}}A [[normal distribution]]<ref>Also called a “[[Gaussian]]” distribution, after the chap who first formulated it, but only by people who are trying to sound clever.</ref> of a series of events, indicates that the events are independent of each other, in that the occurrence of one does not affect the probability of another. [[Coin flip]]s are independent of each other. So are rolls of a die, or the distribution of heights in a classroom. Homo sapiens being the fickle, [[social proof|biddable]] species it is, its cognitive decisions — particularly those concerning [[fashionable idea]]s, to depart quickly from crowded theatres when someone yells fire or to hysterically buy, and then sell, [[Enron]] stock ''[[for fear of missing out]]'' — are not. | ||
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A [[ | |||
===Independent events=== | ===Independent events=== | ||
Independent events fit nicely to a bell curve, as the [[quincunx]] pictured, likes to demonstrate. Bell curves confidently prescribe [[standard deviation]]s, probability intervals, and allow one the comfort to say, “the odds of ''x'' are such that one wouldn’t expect it in several lives of the universe”. When ''x'' really is an independent event (or a series of them) this is prudent enough: “the odds of flipping a coin and getting 99 consecutive heads is ''0.5 x 10<sup>99</sup>'', which you wouldn’t expect in several lifetimes of the universe.” | Independent events fit nicely to a bell curve, as the [[quincunx]] pictured, likes to demonstrate. Bell curves confidently prescribe [[standard deviation]]s, probability intervals, and allow one the comfort to say, “the odds of ''x'' are such that one wouldn’t expect it in several lives of the universe”. When ''x'' really is an independent event (or a series of them) this is prudent enough: “the odds of flipping a coin and getting 99 consecutive heads is ''0.5 x 10<sup>99</sup>'', which you wouldn’t expect in several lifetimes of the universe.” | ||
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Then there are those “[[ten sigma event|ten-sigma” events]] — like, ooooh, say the correlation of a Russian government default with a spike in the price of all other G20 Treasury securities, just to pick something at random — that should, in the world of normal distributions, happen only once in every 10<sup>24</sup> times — say, ten million years — but, since investment decisions are not even remotely independent events, happened once — and only needed to happen once, to blow [[Long Term Capital Management]] and much of the market to smithereens — in ''four'' years. | Then there are those “[[ten sigma event|ten-sigma” events]] — like, ooooh, say the correlation of a Russian government default with a spike in the price of all other G20 Treasury securities, just to pick something at random — that should, in the world of normal distributions, happen only once in every 10<sup>24</sup> times — say, ten million years — but, since investment decisions are not even remotely independent events, happened once — and only needed to happen once, to blow [[Long Term Capital Management]] and much of the market to smithereens — in ''four'' years. | ||
These persist in occurring “against all odds” because they are a product of ''dependent'' events. The distribution of patrons’ arrival times at a cinema are normally distributed around the prescribed showtime because, outside that control, the time at which ''I'' show up has no bearing, or dependency, on the time [[Mrs Pinterman]] shows up. The chance that all 400 people should arrive and try to enter the theatre at the same moment is more or less nil. | These persist in occurring “against all odds” because they are a product of ''dependent'' events. The distribution of patrons’ arrival times at a cinema are normally distributed around the prescribed showtime because, outside that control, the time at which ''I'' show up has no bearing, or dependency, on the time [[Mrs. Pinterman]] shows up. The chance that all 400 people should arrive and try to enter the theatre at the same moment is more or less nil. | ||
But when Mrs. Pinterman then cries “fire” the situational dynamic is very different | But when [[Mrs. Pinterman]] then cries, “fire” the situational dynamic is very different: ''everyone'' tries to leave at once. Even those who didn’t hear Mrs. Pinterman directly, because they instinctively copy everyone else, who did. | ||
When assessing probabilities, therefore, pay attention to the dependency of the events. If events | When assessing probabilities, therefore, pay attention to the dependency of the events. If two events can influence each other — you bought a stock, it went up in price, so I bought it too, kind of thing — ''[[normal distribution]]s do not apply.'' | ||
This seems an obvious lesson; the JC feels less patronising about stating it since failure to heed it led to the collapse of [[LTCM]] ''and'' the [[global financial crisis]]. This from someone who really should have known better: | This seems an obvious lesson; the JC feels less patronising about stating it since failure to heed it led to the collapse of [[LTCM]] ''and'' the [[global financial crisis]]. This from someone who really should have known better: | ||
{{Quote| | {{Quote|{{viniarquote}} | ||
:—David Viniar, Chief Financial Officer, [[Goldman]]}} | :—David Viniar, Chief Financial Officer, [[Goldman]]}} | ||
''Twenty five'' standard deviations. That makes LTCM seem | ''Twenty five'' [[Standard deviation|standard deviations]]. That makes [[LTCM]]’s feeble ''ten'' sigma event seem a virtual certainty. We have it on good authority that the probability of a 25 standard deviation move is 1.309 x 10<sup>130</sup>. <ref>Good [https://www.nottingham.ac.uk/business/who-we-are/centres-and-institutes/gcbfi/documents/cris-reports/cris-paper-2008-3.pdf paper on this from Nottingham University].</ref> That looks a big number, but to a lay person, it doesn’t really have the same impact as writing it out, so let’s do that: | ||
{{Quote|1 day in 1.3 billion billion billion billion billion billion billion billion billion billion billion billion billion billion days. | {{Quote|1 day in 1.3 billion billion billion billion billion billion billion billion billion billion billion billion billion billion days. | ||
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''or'' | ''or'' | ||
1 day in | 1 day in 1 300 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 days.}} | ||
By comparison, the earth is | By comparison, the earth is 1 658 000 000 000 days old, and the universe itself ten times older than that (16 580 000 000 000 000 days). So the [[Goldman]] [[CFO]] was talking about an event that you would only expect once in several trillion trillion trillion trillion lives of the universe, happening ''several days in a row''. | ||
So, no, Mr Viniar: you weren’t seeing cosmos-defying anomalies. ''Your models were wrong''. | |||
But enough already of the chutzpah.<ref>But, [[get your coat]], you know?</ref> The practical lesson is that, unless you are dealing with normally-distributed events, normal probabilities are a ''really'' bad proxy at the extremes. ''Ninety-nine per cent of the way there is nowhere. It isn’t good enough''. | |||
''All'' existential crises sit in the last 1 per cent — last 0.01 per cent, even — because the defining feature of an existential crisis is ''everyone panicking and selling at once''. These are, by definition, the events a normal distribution says will not happen, because events in a normal distribution are independent of each other. | ''All'' existential crises sit in the last 1 per cent — last 0.01 per cent, even — because the defining feature of an existential crisis is ''everyone panicking and selling at once''. These are, by definition, the events a normal distribution says will not happen, because events in a normal distribution are independent of each other. | ||