Quotes About Probability

90 percent of all options expire as losses.

There is actually a law in statistics called the law of iterated expectations, which I outline here in its strong form: if I expect to expect something at some date in the future, then I already expect that something at present.

In the end this is a trivial decision making rule: I am very aggressive when I can gain exposure to positive Black Swans -- when a failure would be of small moment -- and very conservative when I am under threat from a negative Black Swan. I am very aggressive when an error in a model can benefit me, and paranoid when an error can hurt.

investors are led to overestimate their chances of success.

One needs to go out and buy a lottery ticket in order to win. Does it mean that the work involved in the trip to the store caused the winning?

which condition preceded the other.

Fat Tony did not believe in predictions.

Clearly you cannot manufacture more information than the past can deliver; if you buy one hundred copies of The New York Times, I am not too certain that it would help you gain incremental knowledge of the future. We just don't know how much information there is in the past.

Someone who predicts will be fragile to prediction errors.

And numerical prediction leads people to take more risks.

Instead of having medium risk, you have high risk on one side and no risk on the other.

Uncertainty did occasionally pay off there.

should avoid engaging in an action with a big downside if one has no idea of the outcomes.

or mistaking a profitable bet for good forecasting—not convexity of payoff and optionality).

Who is this book written for?

Cygnus Atratus In his Treatise on Human Nature, the Scots philosopher David Hume posed the issue in the following way (as rephrased in the now famous black swan problem by John Stuart Mill): No amount of observations of white swans can allow the inference that all swans are white, but the observation of a single black swan is sufficient to refute that conclusion.

One of the most misunderstood aspects of a Gaussian is its fragility and vulnerability in the estimation of tail events. The odds of a 4 sigma move are twice that of a 4.15 sigma. The odds of a 20 sigma are a trillion times higher than those of a 21 sigma! It means that a small measurement error of the sigma will lead to a massive underestimation of the probability. We can be a trillion times wrong about some events.

the same condition that makes us simplify pushes us to think that the world is less random than it actually is. And the Black Swan is what we leave out of simplification.

do not try to predict precise Black Swans—it tends to make you more vulnerable to the ones you did not predict.

first central assumption leading to the Gaussian bell curve fails in reality. In games, of course, past winnings are not supposed to translate into an increased probability of future gains—but not so in real life, which is why I worry about teaching probability from games. But when winning leads to more winning, you are far more likely to see forty wins in a row than with a proto-Gaussian.

Never cross a river if it is on average four feet deep

Put yourself in situations where favorable consequences are much larger than unfavorable ones.

that which came with the help of luck could be taken away by luck (and often rapidly and unexpectedly at that). The flipside, which deserves to be considered as well (in fact it is even more of our concern), is that things that come with little help from luck are more resistant to randomness.

judged by their actions, not by the results. No matter how sophisticated our choices, how good we are at dominating the odds, randomness will have the last word.