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april 07, 2004

Littlewood's Law of Miracles -The law of truly large numbers

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I The New York Review of Books-recensionen One in a Million recenseras boken Debunked! ESP, Telekinesis, and Other Pseudoscience (bok ej läst). Recensent är Freeman J. Dyson.

The book also has a good chapter on "Amazing Coincidences." These are strange events which appear to give evidence of supernatural influences operating in everyday life. They are not the result of deliberate fraud or trickery, but only of the laws of probability. The paradoxical feature of the laws of probability is that they make unlikely events happen unexpectedly often. A simple way to state the paradox is Littlewood's Law of Miracles. Littlewood was a famous mathematician who was teaching at Cambridge University when I was a student. Being a professional mathematician, he defined miracles precisely before stat-ing his law about them. He defined a miracle as an event that has special significance when it occurs, but oc-curs with a probability of one in a million. This definition agrees with our common-sense understanding of the word "miracle."

Littlewood's Law of Miracles states that in the course of any normal person's life, miracles happen at a rate of roughly one per month. The proof of the law is simple. During the time that we are awake and actively engaged in living our lives, roughly for eight hours each day, we see and hear things happening at a rate of about one per second. So the total number of events that happen to us is about thirty thousand per day, or about a million per month. With few exceptions, these events are not miracles because they are insignificant. The chance of a miracle is about one per million events. Therefore we should expect about one miracle to happen, on the average, every month. Broch tells stories of some amazing coincidences that happened to him and his friends, all of them easily explained as consequences of Littlewood's Law.
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A session with a noticeably high percentage of correct guesses is a miracle according to Littlewood's definition. If a large number of experiments are done by various groups under various conditions, miracles will occasionally occur. If miracles are selectively reported, they are experimentally indistinguishable from real occurrences of telepathy.

Vad jag förstår är det denna lag som även kallas :"The law of truly large numbers". Från Persi Diaconis och Mosteller Methods for Studying Coincidences (min fetning):

Succinctly put, the law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen. The point is that truly rare events, say events that occur only once in a million [as the mathematician Littlewoood (1953) required for an event to be surprising] are bound to be plentiful in a population of 250 million people. If a coincidence occurs to one person in a million each day, then we expect 250 occurences a day and close to 100000 such occurences a year.

Going from year to a lifetime and from the population of the United States to that of the world (5 billion at this writing), we can be absolutely sure that we will see incredibly remarkable events. When such events occur, they are often noted and recorded. If they happen to us or someone we know, it is hard to escape that spooly feeling.

För övrigt nämner Diaconis denna lag i sin On Coincidences-föreläsning.

Se även:
SkepDic: The law of truly large numbers
Littlewoods bok som refereras i Mosteller & Diaconis är A Mathematician's Miscellany (som tydligen heter Littlewood's Miscellany nuförtiden)
John Edensor Littlewood
Quotations by J E Littlewood
samt
Sammanträffanden - anteckningar vid läsning av Diaconis och Mosteller 'Methods for Studying Coincidences'.

Posted by hakank at april 7, 2004 12:59 FM Posted to Sammanträffanden

Comments

Posted by: Carina at april 9, 2004 12:10 FM

Posted by: Håkan Kjellerstrand at april 9, 2004 12:17 FM