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{{a|systems|{{ | {{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. | ||
===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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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 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: | ''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: | ||