Gamblers depicted in "The Flemish Fair". Marten van Cleve. 16th century.

An event is a state of the universe at a particular time in a particular place.


1   Study

The study of the probability of events is called probability. The study of probability begans with Girolamo Cardano’s famed 1564 manuscript De Ludo Aleae, one of the earliest writings on probability and not published until a century after he wrote it, which primarily analyzed dice games. Although Galileo and other 17th-century scientists contributed to this enterprise, many credit the mathematical foundations of probability to an exchange of letters in 1654 between two famous French mathematicians, Blaise Pascal and Pierre de Fermat. They too were concerned with odds and dice throws—for example, whether it is wise to bet even money that a pair of sixes will occur in 24 rolls of two fair dice. [2]

That correspondence inspired significant advances by Abraham de Moivre, Christiaan Huygens, Siméon Poisson, Jacob Bernoulli, Pierre-Simon Laplace and Karl Friedrich Gauss into the 19th century. Still, for a long time there was no formal definition of probability precise enough for use in mathematics or robust enough to handle increasing evidence of random phenomena in science. Not until 1933, nearly four centuries after Cardano, did the Russian mathematician Andrey Kolmogorov put probability theory on a formal axiomatic basis, using the emerging field we now call measure theory. [2]

The history of optimal-stopping problems, a subfield of probability theory, also begins with gambling. One of the earliest discoveries is credited to the eminent English mathematician Arthur Cayley of the University of Cambridge. In 1875, he found an optimal stopping strategy for purchasing lottery tickets. The wider practical applications became apparent gradually. During World War II, Abraham Wald and other mathematicians developed the field of statistical sequential analysis to aid military and industrial decision makers faced with strategic gambles involving massive amounts of men and material. Shortly after the war, Richard Bellman, an applied mathematician, invented dynamic programming to obtain optimal strategies for many other stopping problems. In the 1970s, the theory of optimal stopping emerged as a major tool in finance when Fischer Black and Myron Scholes discovered a pioneering formula for valuing stock options. That transformed the world’s financial markets and won Scholes and colleague Robert Merton the 1997 Nobel Prize in Economics. (Black had died by then.) [2]

The Black-Scholes formula is still the key to modern option pricing, and the optimal-stopping tools underlying it remain a vigorous area of research in academia and industry. But even elementary tools in the theory of optimal stopping offer powerful, practical and sometimes surprising solutions. [2]

1.1   Marriage problem

Suppose you decide to marry, and to select your life partner you will interview at most 100 candidate spouses. The interviews are arranged in random order, and you have no information about candidates you haven’t yet spoken to. After each interview you must either marry that person or forever lose the chance to do so. If you have not married after interviewing candidate 99, you must marry candidate 100. Your objective, of course, is to marry the absolute best candidate of the lot. But how? [2]

I've also put this under search problem. I'm not sure of it.

This problem has a long and rich history in the mathematics literature, where it is known variously as the marriage, secretary, dowry or best-choice problem. Certainly you can select the very best spouse with probability at least 1/100, simply by marrying the first person. But can you do better? [2]

As an enlisted man in the U.S. Air Force during the Vietnam era, John Elton, now a Georgia Institute of Technology mathematician, transformed the marriage problem into a barracks moneymaking scheme. Elton asked his fellow airmen to write down 100 different numbers, positive or negative, as large or small as they wished, on 100 slips of paper, turn them face down on a table and mix them up. He would bet them he could turn the slips of paper over one at a time and stop with the highest number. He convinced them it was “obvious” that the chance of him winning was very small, so he asked ten dollars if he won and paid them one dollar if he lost. There was no shortage of bettors. Even though my friend lost nearly two-thirds of the games, he won more than one-third of them. And with the 10-1 odds, he raked in a bundle. How? [2]

First, note that there is a very simple strategy for winning more than one-fourth of the time, which would already put him ahead. Call an observed number a “record” if it is the highest number seen so far. Suppose you turn over half the numbers—or interview the first 50 marriage candidates—and never stop, no matter how high the number. After that you stop with the first record you see. If the second-highest number in the 100 cards happens to be in the first 50 you look at, and the highest in the second half—which happens 1 in 4 times—then you win. [2]

That strategy is good, but there is an even better one. Observe only 37 cards (or potential partners) without stopping and then stop with the next record. John Gilbert and Frederick Mosteller of Harvard University proved that this strategy is best and guarantees stopping with the best number about 37 percent of the time. In fact, observing N/e ≅ 0.37 of the candidates, where N is the total number of candidates and e is the base of the natural logarithms , e = 2.71828…, guarantees winning with probability more than 1/e > 0.36, no matter how many cards or candidates there are. (Note that the “observe half the numbers” strategy clearly wins with probability at least ¼, also independent of the number of cards.) [2]

Sometimes the goal is to stop with one of the best k of N candidates. That is, you win if you stop with any one of the highest k numbers. In the Olympics or in horse racing, for example, the objective often is the k = 3 case—to win a medal or to show—rather than the all-or-nothing k = 1 goal of a gold medal or a win, which is much riskier. The optimal strategy for stopping with one of the best k is similar to stopping with the best. First, you should observe a fixed number of candidates without ever stopping, thereby obtaining a baseline to work with. Then for another certain fixed length of time, stop if you see a record. Since it is the best seen so far, it is somewhat likely to be one of the best k. If no record appears during that stretch, then continue to the next stage where you stop with one of the highest two numbers for a fixed period of time, and so on. For k = 2, this method guarantees a better than 57 percent chance of stopping with one of the two best even if there are a million cards. For small N , the probability is quite high. [2]

1.2   N = 2

Now, suppose you must decide when to stop and choose between only two slips of paper or two cards. You turn one over, observe a number there and then must judge whether it is larger than the hidden number on the second. The surprising claim, originating with David Blackwell of the University of California, Berkeley, is that you can win at this game more than half the time. Obviously you can win exactly half the time by always stopping with the first number, or always stopping with the second, without even peeking. But to win more than half the time, you must find a way to use information from the first number to decide whether or not to stop. [2]

Here is one stopping rule that guarantees winning more than half the time. First, generate a random number R according to a standard Gaussian (bell-shaped) curve by using a computer or other device. Then turn over one of the slips of paper and observe its number. If R is larger than the observed number, continue and turn over the second card. If R is smaller, quit with the number observed on the first card. [2]

If R is smaller than each of the two written numbers, then you win exactly half the time ( p / 2 of the unknown probability p in Figure 4); if it is larger than both, you again win half that time ( q / 2 of q, also in Figure 4). But if R falls between the two written numbers, which it must do with strictly positive probability (since the two numbers are different and the Gaussian distribution assigns positive probability to every interval) then you win all the time. This gives you the edge you need, since p / 2 + q / 2 + 1–p–q is greater than ½, because 1 - p - q is greater than zero. For example, if the two hidden numbers are 1 and π, this Gaussian method yields a value for p about .8413 and q about .0008, so the probability that it will select the larger number is more than 57 percent. [2]

If the number writer is not completely free to pick any number, but instead is required to choose an integer in the range {1,2,…,100}, say, then he cannot make your probability of winning arbitrarily close to ½. In this case it also seems obvious that the number-writer would never write a 1, since if you turn over a 1, you will always win by not stopping. But if he never writes a 1, he then would never write a 2 either since he never wrote a 1, and so on ad absurdum . Interested readers are invited to discover for themselves the optimal strategy in this case, and the amount more than ½ one can guarantee to win on the average. [2]

1.3   Backward Induction

At the opposite end from having no information about future values is having full information—that is, complete information about the probabilities and the exact values of all potential future observations. In the spirit of Cardano, Fermat and Pascal’s discoveries about probability with dice centuries ago, let’s consider a game of rolling a standard, fair, six-sided die at most five times. You may stop whenever you want and receive as a reward the number of Krugerrands corresponding to the number of dots shown on the die at the time you stop. (At the time of this writing, one Krugerrand is worth $853.87.) Unlike the no-information marriage problems, here everything is known. The values at each roll will be 1, 2, 3, 4, 5, or 6, and the probability of each number on each roll is one-sixth. The objective is to find the stopping rule that will maximize the number of Krugerrands you can expect to win on average. [2]

If you always stop with the first roll, for example, the winnable amount is simply the expected value of a random variable that takes the values 1, 2, 3, 4, 5, and 6 with probability 1/6 each. That is, one-sixth of the time you will win 1, one-sixth of the time you will win 2, and so on, which yields the expected value 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 7/2. Thus if you always quit on the first roll, you expect to win 3.5 Krugerrands on the average. But clearly it is not optimal to stop on the first roll if it is a 1, and it is always optimal to stop with a 6, so already you know part of the optimal stopping rule. Should you stop with a 5 on the first roll? One powerful general technique for solving this type of problem is the method of backward induction. [2]

Clearly it is optimal to stop on the first roll if the value seen on the first roll is greater than the amount expected if you do not stop—that is, if you continue to roll after rejecting the first roll. That would put you in a new game where you are only allowed four rolls, the expected value of which is also unknown at the outset. The optimal strategy in a four-roll problem, in turn, is to stop at the first roll if that value is greater than the amount you expect to win if you continue in a three-roll problem, and so on. Working down, you arrive at one strategy that you do know. In a one-roll problem there is only one strategy, namely to stop, and the expected reward is the expected value of one roll of a fair die, which we saw is 3.5. That information now yields the optimal strategy in a two-roll problem—stop on the first roll if the value is more than you expect to win if you continue, that is, more than 3.5. So now we know the optimal strategy for a two-roll problem—stop at the first roll if it is a 4, 5, or 6, and otherwise continue—and that allows us to calculate the expected reward of the strategy. [2]

In a two-roll problem, you win 4, 5, or 6 on the very first roll, with probability 1/6 each, and stop. Otherwise (the half the time that the first roll was a 1, 2 or 3) you continue, in which case you expect to win 3.5 on the average. Thus the expected reward for the two-roll problem is 4(1/6) + 5(1/6) + 6(1/6) + (1/2)(3.5) = 4.25. This now gives you the optimal strategy for a three-roll problem—namely, stop if the first roll is a 5 or 6 (that is, more than 4.25), otherwise continue and stop only if the second roll is a 4, 5, or 6, and otherwise proceed with the final third roll. Knowing this expected reward for three rolls in turn yields the optimal strategy for a four-roll problem, and so forth. Working backwards, this yields the optimal strategy in the original five-roll problem: Stop on the first roll only if it is a 5 or 6, stop on the second roll if it is a 5 or 6, on the third roll if it is a 5 or 6, the fourth roll if it is a 4, 5 or 6, and otherwise continue to the last roll. This strategy guarantees that you will win about 5.12 Krugerrands on average, and no other strategy is better. (So, in a six-roll game you should stop with the initial roll only if it is a 6.) [2]

In the full-information case, with the objective of stopping with one of the largest k values, the best possible probabilities of winning were unknown for general finite sequences of independent random variables. Using both backward and forward induction, and a class of distributions called “Bernoulli pyramids,” where each new variable is either the best or the worst seen thus far (for example, the first random variable is either +1 or -1 with certain probabilities, the second variable is +2 or -2, and so forth), Douglas Kennedy of the University of Cambridge and I discovered those optimal probabilities. We proved that for every finite sequence of independent random variables, there is always a stop rule that stops with the highest value with probability at least 1/ e , and a stop rule that stops with one of the two highest values with probability at least:

(e ** (-sqrt(2))) * (1 + sqrt(2)) ~= 0.59

2   Form

All events have a cause, a preceding event, and an effect.

An event consists of a location, a start time, and an end time.

3   Properties

3.1   Probability


A random number table.

Probability is a measure of the likeliness that an event will occur. It a measure between zero and one, and is often expressed as a percentage.

Odds are another way of expressing probability, and more applicable to games of chance such as poker. Odds are show as a pair of numbers separated by a colon; the pair represents a ratio between the probability of an event happening and it not happening. For example, saying there is a 70% chance of rain is the same as saying that rain is a 7:3 "favorite" today.

Given that you know rain is a 7:3 favorite, what is a "fair bet"? If you choose to bet on rain and your friend bets on no rain, you should up $7 for each $3 he wagers. Over 10 days, it will probably rain seven times. You will collect $3 from your friend on each rain day for a total of $21; on the remaining three days it will not rain, and your friend will collect a total of $21.

4   Risk

People are risk averse. That is, given a choice between a sure payoff and a probability whose expected value is the same, the majority of people will choose the former.

To offset this effect, risky options need to payoff better. The exact amount depends on a person's willingness to take risk.

Of course, this meets that there is great potential for value. When two people value something differently, trade awaits.

The entrepreneur creates value by taking the risks nobody else would dare to take. Unlike stock traders, to whom risk is a matter of staying in the game so as to be able to wager on future good bets, the entrepreneur is more of an all-or-nothing gamble. As far as transferring risk goes, the entrepreneur is the end of the line.

Venture capital, in that sense, can be seen as in many ways, death insurance. That is to say, while people pay life insurance brokers a constant sum to cover potential accidents, entrepreneurs are paid to share any of their good fortune. [1]

Risk can be reduced with a trial period.

4.1   Causality

Causality is the relation between an event (the cause) and a second event (the effect), where the effect is understood as a consequence of the first.

If an event occurs after an action we may infer that the action probably causes the event.

If no event occurs after an action, we may infer the action probably has no effect.

In either case, we can confirm our inference by repeating the action.

Causality may be direct or indirect. Direct causality is preferred in design.

Eliminate sources of randomness. People are notoriously bad at probability, and seeing patterns that do not exist.

Every time you see a claim like “study links X to Y” or “X is correlated with Y”, you should always always always remember that there are three possibilities: X causes Y, Y causes X, or some third factor Z causes both X and Y. I can’t understate the importance of thinking this way and not jumping to unwarranted conclusions.

4.1.1   Confounding

In statistics, a confounding variable (also confounding factor, hidden variable, lurking variable, a confound, or confounder) is an extraneous variable in a statistical model that correlates (directly or inversely) with both the dependent variable and the independent variable. A perceived relationship between an independent variable and a dependent variable that has been misestimated due to the failure to account for a confounding factor is termed a spurious relationship, and the presence of misestimation for this reason is termed omitted-variable bias.

4.2   History

The philosophical treatment of causality dates back to Aristotle.

Probability theory was developed in the mid-17th century in a fascinating correspondence between two legendary mathematicians, Pierre Fermat (he of Fermat's Last Theorem) and Blaise Pascal (he of Pascal's wager).

4.3   Black Swan

A black swan is:

  1. An outlier
  2. It carries extreme impact
  3. Retrospective predictability: Despite its outlier status, human nature makes us concoct explanations for its occurrence after the fact

A small number of Black Swans explain almost everything in our world.

4.4   Selection bias

During WWII, statistician Abraham Wald was asked to help the British decide where to add armor to their bombers. After analyzing the records, he recommended adding more armor to the places where there was no damage!

This seems backward at first, but Wald realized his data came from bombers that survived. That is, the British were only able to analyze the bombers that returned to England; those that were shot down over enemy territory were not part of their sample. These bombers’ wounds showed where they could afford to be hit. Said another way, the undamaged areas on the survivors showed where the lost planes must have been hit because the planes hit in those areas did not return from their missions.

Wald assumed that the bullets were fired randomly, that no one could accurately aim for a particular part of the bomber. Instead they aimed in the general direction of the plane and sometimes got lucky. So, for example, if Wald saw that more bombers in his sample had bullet holes in the middle of the wings, he did not conclude that Nazis liked to aim for the middle of wings. He assumed that there must have been about as many bombers with bullet holes in every other part of the plane but that those with holes elsewhere were not part of his sample because they had been shot down.

4.5   The Gambler's Ruin

Two gamblers, Alice and Bob, bet on the outcomes of successive flips of a coin. On each flip, if the coin comes up heads, Alice collects $1 from Bob, whereas if it comes up tails, Bob collects $1 from Alice. They continue to do this until one of them runs out of money. If it assumed that the sucessive flips are independent and each flip results in a head with probability p, what is the probability that Alice ends up with all the money if she starts with $A and Bob starts with $B?

Alice will beat Bob with probability 1 – (q/p)^A / ( 1 – (q/p)^(A + B) ) if her probability of winning a flip isn’t 1/2, and simply A / (A + B) if the game is fair.

This is an interesting result with straightforward implications. Since your chance of cleaning the other player out is equal to the amount of money you bring over the total amount of money at the table then if your opponent is a casino you lose. It’s also interesting to note that even if you always bet $2, $5, or any number of dollars for that matter at each flip, the end result is the same.

4.6   Ten hunters

Ten hunters are waiting for ducks to fly by. When a flock of ducks flies overhead, the hunters fire at the same time, but each chooses his target at random, independently of the others. If each hunters independently hits his target with probability p, compute the expected number of ducks that escape unhurt when a flock of size 10 flies overhead.

Let X_i equal 1 if the i-th duck escapes unhurt, and 0 otherwise, for i = 1, 2, ..., 10. The expected number of ducks to escape can be expressed as:

E[X] = E[X_1 + ... + X_10] = E[X_1] + ... + E[X_10]

Each of the hunters will independently hit the i-th duck with probability p/10; the chance that the duck escapes unharmed is equal to the chance that none of the hunters hit it. So:

E[X_i] = P{X_i=1} = (1 - (p / 10)) ^ 10


E[X] = 10 * (1 - (p / 10)) ^ 10.

(Just for reference, if p = 1, E[X] = 3.49.)

If we let T represent the number of targets, and S the number of shooters, then more generally:

E[X] = T * (1 - (p / T)) ^ S

This is pretty cool in my opinion, as the problem shows up a lot in various games (although a lot of games use algorithms to determine which unit to fire at rather than chance), Tower Defense games perhaps being the most obvious.

Above all, chance makes its selection without any recourse to reasons. This quality is perhaps its greatest advantage, though of course it comes at a price. Peter Stone, a political theorist at Trinity College, Dublin, and the author of The Luck of the Draw: The Role of Lotteries in Decision Making (2011), has made a career of studying the conditions under which such reasonless-ness can be, well, reasonable.

‘What lotteries are very good for is for keeping bad reasons out of decisions,’ Stone told me. ‘Lotteries guarantee that when you are choosing at random, there will be no reasons at all for one option rather than another being selected.’ He calls this the sanitising effect of lotteries – they eliminate all reasons from a decision, scrubbing away any kind of unwanted influence. As Stone acknowledges, randomness eliminates good reasons from the running as well as bad ones. He doesn’t advocate using chance indiscriminately. ‘But, sometimes,’ he argues, ‘the danger of bad reasons is bigger than the loss of the possibility of good reasons.’ [1]

Sortition a process of filling certain political positions by lottery.

The practice has a long history. Most public officials in democratic Athens were chosen by lottery, including the nine archons who were chosen by sortition from a significant segment of the population. The nobles of Renaissance Venice used to select their head of state, the doge, through a complicated, partially randomised process. Jean-Jacques Rousseau, in The Social Contract (1762), argued that lotteries would be the norm in an ideal democracy, giving every citizen an equal chance of participating in every part of the government (Rousseau added that such ideal democracies did not exist). Sortition survives today in the process of jury selection, and it crops up from time to time in unexpected places. Ontario and British Columbia, for example, have used randomly selected panels of Canadian citizens to propose election regulations. [1]

Advocates of sortition suggest applying that principle more broadly, to congresses and parliaments, in order to create a legislature that closely reflects the actual composition of a state’s citizenship. They are not (just to be clear) advocating that legislators randomly choose policies. Few, moreover, would suggest that non-representative positions such as the US presidency be appointed by a lottery of all citizens. The idea is not to banish reason from politics altogether. But plenty of bad reasons can influence the election process – through bribery, intimidation, and fraud; through vote-purchasing; through discrimination and prejudices of all kinds. The question is whether these bad reasons outweigh the benefits of a system in which voters pick their favourite candidates. [1]

By way of illustration: a handful of powerful families and influential cliques dominated Renaissance Venice. The use of sortition in selection of the doge, writes the historian Robert Finlay in Politics in Renaissance Venice (1980), was a means of ‘limiting the ability of any group to impose its will without an overwhelming majority or substantial good luck’. Americans who worry about unbridled campaign-spending by a wealthy few might relate to this idea. [1]

Or consider this. In theory, liberal democracies want legislatures that accurately reflect their citizenship. And, presumably, the qualities of a good legislator (intelligence, integrity, experience) aren’t limited to wealthy, straight, white men. The relatively homogeneous composition of our legislatures suggests that less-than-ideal reasons are playing a substantial role in the electoral process. Typically, we just look at this process and wonder how to eliminate that bias. Advocates of sortition see conditions ripe for randomness. [1]

The Swarthmore College professor Barry Schwartz, writing in The Atlantic in 2012, came to a similar conclusion. He proposed that, once schools have

narrowed down their applicant pools to a well-qualified subset, they could just draw names... Once certain standards are met, no really good reasons remain to discriminate between applicant No 2,291 (who gets into Columbia) and applicant No 2,292 (who does not). And once all good reasons are eliminated, the most efficient, most fair and most honest option might be chance. [1]

When randomness is added to a supposedly meritocratic system, it can inspire quite a backlash. In 2004, the International Skating Union (ISU) introduced a new judging system for figure-skating competitions. Under this system – which has since been tweaked – 12 judges evaluated each skater, but only nine of those votes, selected at random, actually counted towards the final tally (the ancient Athenians judged drama competitions in a similar way). Figure skating is a notoriously corrupt sport, with judges sometimes forming blocs that support each other’s favoured skaters. In theory, a randomised process makes it harder to form such alliances. A tit-for-tat arrangement, after all, doesn’t work as well if it’s unclear whether your partners will be able to reciprocate.

But the new ISU rules did more than simply remove a temptation to collude. As statisticians pointed out, random selection will change the outcome of some events. Backing their claims with competition data, they showed how other sets of randomly selected votes would have yielded different results, actually changing the line-up of the medal podium in at least one major competition. Even once all the skaters had performed, ultimate victory depended on the luck of the draw.

There are two ways to look at this kind of situation. The first way – the path of outrage – condemns a system that seems fundamentally unfair. A second approach would be to recognise that the judging process is already subjective and always will be. Had a different panel of 12 judges been chosen for the competition, the result would have varied, too. The ISU system simply makes that subjectivity more apparent, even as it reduces the likelihood that certain obviously bad influences, such as corruption, will affect the final result.

Still, most commentators opted for righteous outrage. That isn’t surprising. The ISU system conflicts with two common modern assumptions: that it is always desirable (and usually possible) to eliminate uncertainty and chance from a situation; and that achievement is perfectly reflective of effort and talent. Sortition, college admission lotteries, and randomised judging run against the grain of both of these premises. They embrace uncertainty as a useful part of their processes, and they fail to guarantee that the better citizen or student or skater, no matter how much she drives herself to success, will be declared the winner.

5   Probability distribution

A probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

5.1   Pareto distribution

A Pareto distribution is ...

In 1906, Italian economist and polymath, Vilfredo Pareto, discovered what became the Pareto principle (= 80-20 rule) when he noticed that 20% of the people owned 80% of the land in Italy - a phenomenon which he found as natural as the fact that 20% of the peapods in his garden produced 80% of the peas. The principle is sometimes known as Price's law, after Derek J. de Sollar Price, the researcher who discovered its application in science in 1963, or the Matthew principle derived a statement by `Jesus Christ`_ (Matthew 25:29) "to those who have everything, more will be given; from those who have nothing, everything will be taken". The principle describes many natural and artificial systems, including the population of cities (a small number have almost all the people), the mass of heavenly bodies (a small number hoard all the matter), and the frequency of words (90 percent of communication occurs using just 500 words). [4]

6   Further reading

7   References

[1](1, 2, 3, 4, 5, 6)
Michael Schulson. How to choose?


[2](1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19) Theodore Hill. Knowing When to Stop. http://www.americanscientist.org/issues/id.5783,y.2009,no.2,content.true,page.1,css.print/issue.aspx
[3]Regina Nuzzo. Feb 12, 2014. Scientific method: Statistical errors. http://www.nature.com/news/scientific-method-statistical-errors-1.14700
[4]Jordan B. Peterson. Jan 16, 2018. 12 Rules for Life: An Antidote to Chaos.

My thoughts on Hindsight bias

Now, at the end though, I feel really confused how retrospect works. In other words, how is it that we can have events A, B, C, and D, and not see that they "should cause" E, but when E happens, it seems obvious that they should.

It seems like a better analogy than beads might be receiving cards from the dealer during a round of poker. You get a 10 and it's interesting, but on its own it doesn't mean anything. Then a Jack, a King, a Ace-- all the same. Then suddenly a Queen! And now everything matters.

In the card analogy, of course receiving the 10 at the start doesn't cause the Queen at the end. And if you agree with my analogy, I think your conclusions seem funny:

> 1. Do not to despair at reading these stories. They are excellent inspirational material, but poor instructional manuals.

Very true. And it's funny that that we ever though they could be.

> 2. Even though we see it in movies and read about it in magazines it’s not necessary to imbue forced significance on to any particular event or idea as it happens – many successful people didn’t recognize the importance of their experiences until many years after they occurred.

Also very true-- nor could they.

> 3. Whether we recognize it or not the potential for something extraordinary lies in pretty much every moment of our lives, large or small... And the beauty of it is this: the not-knowing what our story will be requires faith that eventually the beads of our lives will be connected on a string. And it encourages a sense of wonder as we look at every moment – because every moment could be one that becomes woven into the fabric of our life’s story...

True, but it doesn't mean we should celebrate each event, just like you wouldn't celebrate the first 10.

Of course, the analogy of receiving cards isn't exactly perfect-- in life you can influence what you're given. I wonder if instead of having faith that the next card will be the one I need, I could look at my hand and try to make it so.

Monty Hall Problem.

Case 1 - Pick prize
Switch - Lose Stay - Win
Case 2 - Pick loss
Switch - Win Stay - Lose
Case 3 - Pick loss
Switch - Win Stay - Lose

If you stay, 1/3 chance of winning. If you switch, 2/3 chance of winning.

The lottery organizers base their work on a well-understood facet of human nature: people don't understand odds. They pay less attention to the probability of a given event than the consequences if it takes place. The odds of a terrorist incident on a plane come to 1 per 16.6 million departures, statistics guru Nate Silver calculated in 2009, but people tend to focus on the outcome -- hundreds of passengers perishing at once -- than on the probabilities.



The odds of being struck by lightning this year are one in 1.19 million, making it about 246 times as likely as winning the Powerball jackpot.

With an estimated one in 12,500 chance, an amateur golfer is about 23,376 times as likely to make a hole in one.


K statistician Ronald Fisher introduced the P value in the 1920s. He intended it simply as an informal way to judge whether evidence was significant in the old-fashioned sense: worthy of a second look. The idea was to run an experiment, then see if the results were consistent with what random chance might produce.

Researchers would first set up a 'null hypothesis' that they wanted to disprove, such as there being no correlation or no difference between two groups. Next, they would play the devil's advocate and, assuming that this null hypothesis was in fact true, calculate the chances of getting results at least as extreme as what was actually observed. This probability was the P value. The smaller it was, suggested Fisher, the greater the likelihood that the straw-man null hypothesis was false.

Soon it got swept into a movement to make evidence-based decision-making as rigorous and objective as possible. This movement was spearheaded in the late 1920s by Fisher's bitter rivals, Polish mathematician Jerzy Neyman and UK statistician Egon Pearson, who introduced an alternative framework for data analysis that included statistical power, false positives, false negatives and many other concepts now familiar from introductory statistics classes. They pointedly left out the P value.

Other researchers lost patience and began to write statistics manuals for working scientists. And because many of the authors were non-statisticians without a thorough understanding of either approach, they created a hybrid system that crammed Fisher's easy-to-calculate P value into Neyman and Pearson's reassuringly rigorous rule-based system.

Critics also bemoan the way that P values can encourage muddled thinking. A prime example is their tendency to deflect attention from the actual size of an effect. Last year, for example, a study of more than 19,000 people showed8 that those who meet their spouses online are less likely to divorce (p < 0.002) and more likely to have high marital satisfaction (p < 0.001) than those who meet offline (see Nature http://doi.org/rcg; 2013). That might have sounded impressive, but the effects were actually tiny: meeting online nudged the divorce rate from 7.67% down to 5.96%, and barely budged happiness from 5.48 to 5.64 on a 7-point scale... But significance is no indicator of practical relevance, he says: “We should be asking, 'How much of an effect is there?', not 'Is there an effect?'”

Some studies get published with no peer review at all, as so-called “predatory publishers” flood the scientific literature with journals that are essentially fake, publishing any author who pays. Jeffrey Beall, a librarian at the University of Colorado at Denver, has compiled a list of more than 100 so-called “predatory” journal publishers. These journals often have legit-sounding names like the International Journal of Advanced Chemical Research and create opportunities for crackpots to give their unscientific views a veneer of legitimacy. (The fake “get me off your fucking mailing list” and “Simpsons” papers were published in such journals.)

Predatory journals flourish, in part, because of the sway that publication records have when it comes to landing jobs and grants, creating incentives for researchers to pad their CVs with extra papers.

Gambling has spawned more than entertainment and individual profits and losses over the millennia. The drive to improve the odds of winning gave birth to the mathematical field of probability, which in turn produced optimal stopping strategies. These strategies can improve one’s odds of making a good choice and are useful in many more decisions than those demanded in games of chance. [2]

A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG),[1] is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers