"A normal distribution is most unlikely, although not
impossible, when the observations are dependent upon one another - that is, when
the probability of one event is determined by a preceding event. The
observations will fail to distribute themselves symmetrically around the mean."
"All the law [of large numbers] tells us is that the average of a large number
of throws will be more likely than the average of a small number of throws to
differ from the true average by less than some stated amount. And there will
always be a possibility that the observed result will differ from the true
average by a larger amount than the specified bound."
"But real-life situations often require us to measure probability in precisely this fashion - from sample to universe. In only rare cases does life replicate games of chance, for which we can determine the probability of an outcome before an event even occurs - a priori […] . In most instances, we have to estimate probabilities from what happened after the fact - a posteriori. The very notion of a posteriori implies experimentation and changing degrees of belief."
"In a mathematical sense a zero-sum game is a loser's game when
it is valued in terms of utility. The best decision for both is to refuse to
play this game."
"Jacob Bernoulli's theorem for calculating probabilities a posteriori is known as the Law of Large Numbers. Contrary to the popular view, this law does not provide a method for validating observed facts, which are only an incomplete representation of the whole truth. Nor does it say that an increasing number of observations will increase the probability that what you see is what you are going to get. The law is not a design for improving the quality of empirical tests […]." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"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)
"Pascal's Triangle and all the early work in probability answered only one question: what is the probability of such-and-such an outcome? The answer to that question has limited value in most cases, because it leaves us with no sense of generality." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"Probability has always carried this double meaning, one
looking into the future, the other interpreting the past, one concerned with
our opinions, the other concerned with what we actually know."
"Probability theory is a serious instrument for forecasting,
but the devil, as they say, is in the details - in the quality of information
that forms the basis of probability estimates.
"Reality is a series of connected events, each dependent on another, radically different from games of chance in which the outcome of any single throw has zero influence on the outcome of the next throw. Games of chance reduce everything to a hard number, but in real life we use such measures as 'a little', 'a lot', or 'not too much, please' much more often than we use a precise quantitative measure." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"So we pour in data from the past to fuel the decision-making mechanisms created by our models, be they linear or nonlinear. But therein lies the logician's trap: past data from real life constitute a sequence of events rather than a set of independent observations, which is what the laws of probability demand. [...] It is in those outliers and imperfections that the wildness lurks." (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)
"The Law of Large Numbers does not tell you that the average of your throws will approach 50% as you increase the number of throws; simple mathematics can tell you that, sparing you the tedious business of tossing the coin over and over. Rather, the law states that increasing the number of throws will correspondingly increase the probability that the ratio of heads thrown to total throws will vary from 50% by less than some stated amount, no matter how small. The word 'vary' is what matters. The search is not for the true mean of 50% but for the probability that the error between the observed average and the true average will be less than, say, 2% - in other words, that increasing the number of throws will increase the probability that the observed average will fall within 2% of the true average." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"The resolution of how to divide the stakes in an uncompleted game marked the beginning of a systematic analysis of probability - the measure of our confidence that something is going to happen. It brings us to the threshold of the quantification of risk." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"The theory of probability can define the probabilities at
the gaming casino or in a lottery - there is no need to spin the roulette wheel
or count the lottery tickets to estimate the nature of the outcome - but in real
life relevant information is essential. And the bother is that we never have
all the information we would like. Nature has established patterns, but only
for the most part. Theory, which abstracts from nature, is kinder: we either
have the information we need or else we have no need for information."
"Time is the dominant factor in gambling. Risk and time are opposite sides of the same coin, for if there were no tomorrow there would be no risk. Time transforms risk, and the nature of risk is shaped by the time horizon: the future is the playing field." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"Under conditions of uncertainty, both rationality and measurement are essential to decision-making. Rational people process information objectively: whatever errors they make in forecasting the future are random errors rather than the result of a stubborn bias toward either optimism or pessimism. They respond to new information on the basis of a clearly defined set of preferences. They know what they want, and they use the information in ways that support their preferences." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"Under conditions of uncertainty, the choice is not between rejecting a hypothesis and accepting it, but between reject and not - reject. You can decide that the probability that you are wrong is so small that you should not reject the hypothesis. You can decide that the probability that you are wrong is so large that you should reject the hypothesis. But with any probability short of zero that you are wrong - certainty rather than uncertainty - you cannot accept a hypothesis." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"Until we can distinguish between an event that is truly
random and an event that is the result of cause and effect, we will never know whether
what we see is what we'll get, nor how we got what we got. When we take a risk,
we are betting on an outcome that will result from a decision we have made,
though we do not know for certain what the outcome will be. The essence of risk
management lies in maximizing the areas where we have some control over the
outcome while minimizing the areas where we have absolutely no control over the
outcome and the linkage between effect and cause is hidden from us."
"We can assemble big pieces of information and little pieces, but we can never get all the pieces together. We never know for sure how good our sample is. That uncertainty is what makes arriving at judgments so difficult and acting on them so risky. […] When information is lacking, we have to fall back on inductive reasoning and try to guess the odds." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"Whenever we make any decision based on the expectation that matters will return to 'normal', we are employing the notion of regression to the mean." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
"Without numbers, there are no odds and no probabilities; without odds and probabilities, the only way to deal with risk is to appeal to the gods and the fates. Without numbers, risk is wholly a matter of gut." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)
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