William Feller: Number

“Simple methods will soon lead us to results of far reaching theoretical and practical importance. We shall encounter theoretical conclusions which not only are unexpected but actually come as a shock to intuition and common sense. They will reveal that commonly accepted notions concerning chance fluctuations are without foundation and that the implications of the law of large numbers are widely misconstrued.”

William Feller

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter III, Fluctuations In Coin Tossing And Random Walks, p. 67.

Law , Sense , Chance

“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

“It is seen that continued shuffling may reasonably be expected to produce perfect "randomness" and to eliminate all traces of the original order. It should be noted, however, that the number of operations required for this purpose is extremely large.”

William Feller

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter XV, Markov Chains, p. 407.

Perfection

“This faulty intuition as well as many modern applications of probability theory are under the strong influence of traditional misconceptions concerning the meaning of the law of large numbers and of a popular mystique concerning a so-called law of averages.”

William Feller

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter X, Law Of large Numbers, p. 250.

Law , Strong

“It is a common fallacy to believe that the law of large numbers acts as a force endowed with memory seeking to return to the original state, and many wrong conclusions have been drawn from this assumption.”

William Feller

Source: An Introduction To Probability Theory And Its Applications (Third Edition), Chapter V, Conditional Probability, Stochastic Independence, p. 136.

Law , Memory