"Equiprobability in the physical world is purely a hypothesis. We may exercise the greatest care and the most accurate of scientific instruments to determine whether or not a penny is symmetrical. Even if we are satisfied that it is, and that our evidence on that point is conclusive, our knowledge, or rather our ignorance, about the vast number of other causes which affect the fall of the penny is so abysmal that the fact of the penny’s symmetry is a mere detail. Thus, the statement 'head and tail are equiprobable' is at best an assumption." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"A misunderstanding of Bernoulli’s theorem is responsible for one of the commonest fallacies in the estimation of probabilities, the fallacy of the maturity of chances. When a coin has come down heads twice in succession, gamblers sometimes say that it is more likely to come down tails next time because ‘by the law of averages’ (whatever that may mean) the proportion of tails must be brought right some time." (William Kneale, "Probability and Induction", 1949)
"We must emphasize that such terms as 'select at random', 'choose at random', and the like, always mean that some mechanical device, such as coins, cards, dice, or tables of random numbers, is used." (Frederick Mosteller et al, "Principles of Sampling", Journal of the American Statistical Association Vol. 49 (265), 1954)
"And nobody can get [...] far without at least an acquaintance with the mathematics of probability, not to the extent of making its calculations and filling examination papers with typical equations, but enough to know when they can be trusted, and when they are cooked. For when their imaginary numbers correspond to exact quantities of hard coins unalterably stamped with heads and tails, they are safe within certain limits; for here we have solid certainty [...] but when the calculation is one of no constant and several very capricious variables, guesswork, personal bias, and pecuniary interests, come in so strong that those who began by ignorantly imagining that statistics cannot lie end by imagining equally ignorantly, that they never do anything else." (George B Shaw, "The World of Mathematics", 1956)
"[...] there can be such a thing as a simple probabilistic system. For example, consider the tossing of a penny. Here is a perfectly simple system, but one which is notoriously unpredictable. It maybe described in terms of a binary decision process, with a built-in even probability between the two possible outcomes." (Stafford Beer, "Cybernetics and Management", 1959)
"The shrewd guess, the fertile hypothesis, the courageous leap to a tentative conclusion - these are the most valuable coin of the thinker at work." (Jerome S Bruner, "The Process of Education", 1960)
"No Chancellor of the Exchequer could introduce his proposals for monetary and fiscal policy in the House of Commons by saying 'I have looked at all the forecasts, some go one way, some another; so I decided to toss a coin and assume inflationary tendencies if it came down heads and deflationary if it came down tails' [...] And statistics, however uncertain, can apparently provide some basis." (Ely Devons, "Essays in Economics", 1961)
"The equanimity of your average tosser of coins depends upon a law, or rather a tendency, or let us say a probability, or at any rate a mathematically calculable chance, which ensures that he will not upset himself by losing too much nor upset his opponent by winning too often." (Tom Stoppard, "Rosencrantz and Guildenstern Are Dead", 1967)
"A significant property of the value function, called loss aversion, is that the response to losses is more extreme than the response to gains. The common reluctance to accept a fair bet on the toss of a coin suggests that the displeasure of losing a sum of money exceeds the pleasure of winning the same amount. Thus the proposed value function is (i) defined on gains and losses, (ii) generally concave for gains and convex for losses, and (iii) steeper for losses than for gains."
"Flip a coin 100 times. Assume that 99 heads are obtained. If you ask a statistician, the response is likely to be: 'It is a biased coin'. But if you ask a probabilist, he may say: 'Wooow, what a rare event'." (Chamont Wang, "Sense and Nonsense of Statistical Inference", 1993)
"The coin is an example of complete randomness. It is the sort of randomness that one commonly has in mind when thinking of random numbers, or deciding to use a random-number generator." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Losing streaks and winning streaks occur frequently in games of chance, as they do in real life. Gamblers respond to these events in asymmetric fashion: they appeal to the law of averages to bring losing streaks to a speedy end. And they appeal to that same law of averages to suspend itself so that winning streaks will go on and on. The law of averages hears neither appeal. The last sequence of throws of the dice conveys absolutely no information about what the next throw will bring. Cards, coins, dice, and roulette wheels have no memory." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"The dice and the roulette wheel, along with the stock market and the bond market, are natural laboratories for the study of risk because they lend themselves so readily to quantification; their language is the language of numbers." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"However, random walk theory also tells us that the chance that the balance never returns to zero - that is, that H stays in the lead for ever - is 0. This is the sense in which the 'law of averages' is true. If you wait long enough, then almost surely the numbers of heads and tails will even out. But this fact carries no implications about improving your chances of winning, if you're betting on whether H or T turns up. The probabilities are unchanged, and you don't know how long the 'long run' is going to be. Usually it is very long indeed." (Ian Stewart, The Magical Maze: Seeing the world through mathematical eyes", 1997)
"In everyday language, a fair coin is called random, but not a coin that shows head more often than tail. A coin that keeps a memory of its own record of heads and tails is viewed as even less random. This mental picture is present in the term random walk, especially as used in finance." (Benoit B Mandelbrot, "Fractals and Scaling in Finance: Discontinuity, concentration, risk", 1997)
"The basis of many misconceptions about probability is a belief in something usually referred to as 'the law of averages', which alleges that any unevenness in random events gets ironed out in the long run. For example, if a tossed coin keeps coming up heads, then it is widely believed that at some stage there will be a predominance of tails to balance things out." (Ian Stewart, The Magical Maze: Seeing the world through mathematical eyes", 1997)
"Sequences of random numbers also inevitably display certain regularities. […] The trouble is, just as no real die, coin, or roulette wheel is ever likely to be perfectly fair, no numerical recipe produces truly random numbers. The mere existence of a formula suggests some sort of predictability or pattern." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)
"If sinks, sources, saddles, and limit cycles are coins landing heads or tails, then the exceptions are a coin landing on edge. Yes, it might happen, in theory; but no, it doesn't, in practice." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)
"It's a bit like having a theory about coins that move in space, but only being able to measure their state by interrupting them with a table. We hypothesize that the coin may be able to revolve in space, a state that is neither ‘heads’ nor ‘tails’ but a kind of mixture. Our experimental proof is that when you stick a table in, you get heads half the time and tails the other half - randomly. This is by no means a perfect analogy with standard quantum theory - a revolving coin is not exactly in a superposition of heads and tails - but it captures some of the flavour." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)
"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing – seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)
"The possibility of translating uncertainties into risks is much more restricted in the propensity view. Propensities are properties of an object, such as the physical symmetry of a die. If a die is constructed to be perfectly symmetrical, then the probability of rolling a six is 1 in 6. The reference to a physical design, mechanism, or trait that determines the risk of an event is the essence of the propensity interpretation of probability. Note how propensity differs from the subjective interpretation: It is not sufficient that someone’s subjective probabilities about the outcomes of a die roll are coherent, that is, that they satisfy the laws of probability. What matters is the die’s design. If the design is not known, there are no probabilities." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)
"Randomness is a difficult notion for people to accept. When events come in clusters and streaks, people look for explanations and patterns. They refuse to believe that such patterns - which frequently occur in random data - could equally well be derived from tossing a coin. So it is in the stock market as well." (Didier Sornette, "Why Stock Markets Crash: Critical events in complex financial systems", 2003)
"Suppose that while flipping a coin, a small black hole passed by and ate the coin. As long as we got to see the coin, the probabilities of heads and tails would add to one, but the possibility of a coin disappearing altogether into a black hole would have to be included. Once the coin crosses the event horizon of the black hole, it simply does not meaningfully exist in our universe anymore. Can we simply adjust our probabilistic interpretation to accommodate this outcome? Will we ever encounter negative probabilities?"
"A bit involves both probability and an experiment that decides a binary or yes-no question. Consider flipping a coin. One bit of in-formation is what we learn from the flip of a fair coin. With an unfair or biased coin the odds are other than even because either heads or tails is more likely to appear after the flip. We learn less from flipping the biased coin because there is less surprise in the outcome on average. Shannon's bit-based concept of entropy is just the average information of the experiment. What we gain in information from the coin flip we lose in uncertainty or entropy." (Bart Kosko, "Noise", 2006)
"Random number generators do not always need to be symmetrical. This misconception of assuming equal likelihood for each outcome is fostered in a restricted learning environment, where learners see only such situations (that is, dice, coins and spinners). It is therefore very important for learners to be aware of situations where the different outcomes are not equally likely (as with the drawing-pins example).
"The objectivist view is that probabilities are real aspects of the universe - propensities of objects to behave in certain ways - rather than being just descriptions of an observer’s degree of belief. For example, the fact that a fair coin comes up heads with probability 0.5 is a propensity of the coin itself. In this view, frequentist measurements are attempts to observe these propensities. Most physicists agree that quantum phenomena are objectively probabilistic, but uncertainty at the macroscopic scale - e.g., in coin tossing - usually arises from ignorance of initial conditions and does not seem consistent with the propensity view." (Stuart J Russell & Peter Norvig, "Artificial Intelligence: A Modern Approach", 2010)
"A very different - and very incorrect - argument is that successes must be balanced by failures (and failures by successes) so that things average out. Every coin flip that lands heads makes tails more likely. Every red at roulette makes black more likely. […] These beliefs are all incorrect. Good luck will certainly not continue indefinitely, but do not assume that good luck makes bad luck more likely, or vice versa."
"Remember that even random coin flips can yield striking, even stunning, patterns that mean nothing at all. When someone shows you a pattern, no matter how impressive the person’s credentials, consider the possibility that the pattern is just a coincidence. Ask why, not what. No matter what the pattern, the question is: Why should we expect to find this pattern?"
"We are seduced by patterns and we want explanations for these patterns. When we see a string of successes, we think that a hot hand has made success more likely. If we see a string of failures, we think a cold hand has made failure more likely. It is easy to dismiss such theories when they involve coin flips, but it is not so easy with humans. We surely have emotions and ailments that can cause our abilities to go up and down. The question is whether these fluctuations are important or trivial."
"When statisticians, trained in math and probability theory, try to assess likely outcomes, they demand a plethora of data points. Even then, they recognize that unless it’s a very simple and controlled action such as flipping a coin, unforeseen variables can exert significant influence." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)
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