William Feller: Coin

“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.

Game , Chance , Time

“It should be noted that this duration is considerably longer than we naively expect. If two players with 500 dollars each toss a coin until one is ruined, the average duration of the game is 250,000 trials. If a gambler has only one dollar and his adversary has 1000, the average duration is 1000 trials.”

William Feller

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter XIV, Random Walk And Ruin Problems, p. 349.

Game

“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

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

Law , Game , Time , Reading

“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

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

Law

“The man in the street, and also the philosopher K. Marbe, believe that after a run of seventeen heads tail becomes more probable. This argument has nothing to do with imperfections of physical coins; it endows nature with memory, or, in our terminology, it denies the stochastic independence of successive trials. Marbe's theory cannot be refuted by logic but is rejected because of empirical support.”

William Feller

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter VI, The Binomial And The Poisson Distributions, p. 147.

Support , Nature , Memory , Success

“It has been suggested that an army of monkeys might be trained to pound typewriters at random in the hope that ultimately great works of literature would be produced. Using a coin for the same purpose may save feeding and training expenses and free the monkeys for other monkey business.”

William Feller

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter VIII, Unlimited Sequences Of Bernoulli Trials, p. 202.

Hope , Literature , Business , Training

“When a coin is tossed, it does not necessarily fall heads or tails; it can roll away or stand on its edge.”

William Feller

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter I, The Sample Space, p. 7