“The fact that the mean recurrence time is infinite implies that the chance fluctuations in an individual prolonged coin-tossing game are far removed from the familiar pattern governed by the normal distribution.”

—  William Feller

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter XIII, Recurrent Events. Renewal Theory. p. 314.

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "The fact that the mean recurrence time is infinite implies that the chance fluctuations in an individual prolonged coin…" by William Feller?
William Feller photo
William Feller30
Croatian-American mathematician 1906–1970

Related quotes

“Warning. It is usual to read into the law of large numbers things which it definitely does not imply. If Peter and Paul toss a perfect coin 10,000 times, it is customary to expect that Peter will be in the lead roughly half the time. This is not true. In a large number of different coin-tossing games it is reasonable to expect that any fixed moment heads will be in the lead in roughly half of all cases. But it is quiet likely that the player who ends at the winning side has been in the lead for practically the whole duration of the game. Thus, contrary to widespread belief, the time average for any individual game has nothing to do with the ensemble average at any given moment.”

William Feller (1906–1970) Croatian-American mathematician

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter VI, The Binomial And The Poisson Distributions, p. 152-153.

Karl Pearson photo

“While the moralists have boldly asserted that the service of the goddess Chance leads to a complete demoralisation of her worshippers... the mathematicians... start from the hypothesis that gambling is a game of chance—chance being defined in their own perfectly clear and definite sense. My object... is to show that chance in this sense... as it applies to the tossing of an unloaded coin, has no application to Monte Carlo roulette.”

Karl Pearson (1857–1936) English mathematician and biometrician

"The Scientific Aspect of Monte Carlo Roulette" (1894)

“The results concerning fluctuations in coin tossing show the widely held beliefs about the law of large numbers are fallacious. They were so amazing and so at variance with common intuition that even sophisticated colleagues doubted that coins actually misbehave as theory predicts.”

William Feller (1906–1970) Croatian-American mathematician

Note on the Use of this Book, p. xi-xii.
An Introduction To Probability Theory And Its Applications (Third Edition)

“In American culture the coin toss is the paradigm of the random event. A coin toss decides who kicks off the Super Bowl. Looked at another way, a coin toss is not random at all. It is physics.”

William Poundstone (1955) American writer

Part One, Entropy, Toy Room, p. 46
Fortune's Formula (2005)

“The pattern would have been familiar—bohemian, creative, arty—except that it was even further removed from reality, Romanticism in its furthest decadence; being only an exhausted impersonation of poverty, rebellion and artistic “soul.” For it was the unhappy fact that most of them worked for a living and obtained the substance of their conversation from the pages of Time magazine and like publications”

Thomas Pynchon book V.

Source: V. (1963), Chapter Two, Part II
Context: The rest of the Crew partook of the same lethargy. Raoul wrote for television, keeping carefully in mind, and complaining bitterly about, all the sponsor-fetishes of that industry. Slab painted in sporadic bursts, referring to himself as a Catatonic Expressionist and his work as “the ultimate in non-communication.” Melvin played the guitar and sang liberal folk songs. The pattern would have been familiar—bohemian, creative, arty—except that it was even further removed from reality, Romanticism in its furthest decadence; being only an exhausted impersonation of poverty, rebellion and artistic “soul.” For it was the unhappy fact that most of them worked for a living and obtained the substance of their conversation from the pages of Time magazine and like publications

Rose Wilder Lane photo

“In actual fact, the population of a country is a multitude of diverse human beings with an infinite variety of purposes and desires and fluctuating wills”

Rose Wilder Lane (1886–1968) American journalist

Give Me Liberty (1936)
Context: Representative government cannot express the will of the mass of the people, because there is no mass of the people; The People is a fiction, like The State. You cannot get a Will of the Mass, even among a dozen persons who all want to go on a picnic. The only human mass with a common will is a mob, and that will is a temporary insanity. In actual fact, the population of a country is a multitude of diverse human beings with an infinite variety of purposes and desires and fluctuating wills.

Michael Crichton photo

“You could not predict what would happen in a single instance, a single throw of the dice, a single pitch in the seventh inning, a single toss of the coin. But you could predict three out of five, four out of ten, seven out of sixteen, and to that extent chance governed everyone, all the time. Just as surely as two equals two.”

Michael Crichton book Odds On

Odds On (1966)

“Our life is one, big proverbial coin toss.”

L.J. Smith (1965) American author

Michael Crichton photo

“In the conservative region far from the chaotic edge, individual elements coalesce slowly, showing no clear pattern.”

Michael Crichton book The Lost World

Source: First Configuration, The Lost World (1995)

Huston Smith photo

“From a purely human standpoint the wisdom traditions are the species' most prolonged and serious attempts to infer from the maze on this side of the tapestry the pattern which, on its right side, gives meaning to the whole.”

Huston Smith book The World's Religions

The World's Religions (1991)
Context: The religions begin by assuring us that if we could see the full picture we would find it more integrated than we would normally suppose. Life gives us no view of the whole. [... ] It is as if life were a great tapestry, which we face from its wrong side. This gives it the appearance of a maze of knots and threads, which for the most part appear chaotic.
From a purely human standpoint the wisdom traditions are the species' most prolonged and serious attempts to infer from the maze on this side of the tapestry the pattern which, on its right side, gives meaning to the whole. As the beauty and harmony of the design derive from the way its parts are related, the design confers on these parts a significance that we, seeing only scraps of the design, do not normally perceive.

Related topics