Innumeracy by John Allen Paulos

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The Notes

• “Mathematical solecisms of one form or another are a bit like piles of garbage; no matter how often they’re picked up, they soon collect again.”
• “…a truism: that numbers and statistics always require interpretation.”
• “Innumeracy, an inability to deal comfortably with the fundamental notions of number and chance, plagues far too many otherwise knowledgeable citizens.”
• “Innumerate people characteristically have a strong tendency to personalize—to be misled by their own experiences, or by the media’s focus on individuals and drama.
• The real-world consequences of innumeracy open people up to pseudoscience, stock scams, wild medical and diet claims, conspiracies, misperceived risks, and more.
• People have a poor grasp of just how large, big numbers are and how tiny, small numbers can be.
• That leads to a poor understanding of probabilities which leads to overweighting unlikely risks and underweighting common risks. It also leads to focus more on rare outcomes and mistakenly believing rare outcomes are more likely than they actually are.
• People tend to personalize risks, relying on anecdotal experience rather than statistics.
• Big Numbers in Context
  • 1 million seconds is about 11.5 days
  • 1 billion seconds is about 32 years
  • 1 trillion seconds is about 32,000 years
  • The best way to wrap your mind around huge numbers is to have familiar examples of corresponding smaller numbers — of 1,000 or 10,000 — that be taken to a power of 10 or more to better understand the magnitude of something.
• “If people were more capable of estimation and simple calculation, many obvious inferences would be drawn (or not), and fewer ridiculous notions would be entertained.”
• People have a poor understanding of how small quantities can add up to a massive impact over time.
  • One example is misunderstanding the power the compounding.
  • Archimedes’ fulcrum is another: given a long enough lever, he could lift the Earth.
• Multiplication Principle = solves for the number of possible outcomes across two or more events by multiplying the number of possible outcomes for each event.
  • Possible outcomes rolling a pair of dice = 6 x 6 = 36
  • Possible outcomes if the second dice rolled is different from the first = 6 x 5 = 30. (It’s 6 x 5 because the second dice must exclude the outcome of the first dice).
  • Number of combinations of triple scoop cones without flavor repetition at Baskin Robbins (31 flavors) = 31 x 30 x 29 = 26,970.
  • Number of cones without flavor repetition regardless of the order of flavor = 26,970 /6 = 4,495 cones. (where 6 = 3 x 2 x 1 or the number of ways to arrange three flavors). This is an example of combinatorial coefficient.
  • Combinatorial Coefficient = a way to solve for X elements out of Y elements regardless of the order of X elements chosen
• The probability of two independent events occurring is solved with the multiplication principle.
  • The probability of two consecutive coin flips coming up heads is 1/2 x 1/2 = 1/4 or 25%.
  • The probability of five consecutive coin flips coming up heads is 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32 or 3.125%
  • The probability that an event does not occur is 1 minus the probability that it does occur.
• “A tendency to drastically underestimate the frequency of coincidences is a prime characteristic of innumerates, who generally accord great significance to correspondences of all sorts while attributing too little significance to quite conclusive but less flashy statistical evidence.”
• “Few experiences are more dispiriting to me than meeting someone who seems intelligent and open to the world but who immediately inquires about my zodiac sign and then begins to note characteristics of my personality consistent with that sign (whatever sign I give them).”
• Coincidences are often just more likely to happen than we want to believe.
  • The odds that 2 or more people, out of a group of 23 random people, share any birthday is 50%.
  • The odds that 2 or more people, out of a group of 253 random people, share a specific birthday (like April 12), is 50%.
  • Chance encounters are mathematically common.
    • The odds that 2 random people are linked through two acquaintances is about 99% (Like 6 degrees of Kevin Bacon).
    • There’s a 63% chance that given two decks of cards, at least one exact match will occur if cards from both decks are turned over one at a time.
• Predictions
  • “The paradoxical conclusion is that it would be very unlikely for unlikely events not to occur. If you don’t specify a predicted event precisely, there are an indeterminate number of ways for an event of that general kind to take place.”
  • “…predictions are usually sufficiently vague so that the probability of some event of the predicted kind occurring is very high; it’s the particular predictions that seldom come true.”
• Stock Newsletter Scam
  • Send newsletter predicting the direction of a stock or index to 16,000 investors. Half of those sent predict a rise. Half predict a fall.
  • The next newsletter is sent only to those that received the correct “prediction” with another prediction. Again, half of those sent predict a rise. Half predict a fall.
  • This is repeated until the remaining 500 investor received correct prediction six times.
  • To get the seventh, they must pay a high-priced subscription fee.
• Similar to the newsletter scam is the tendency for investors to stay quiet about losing strategies and boast winning strategies. This can cause strategies to “work out” for a while as more people pile in despite being baseless.
  • The media creates a similar dynamic by writing up big winners or stocks that have run up a lot and ignoring losers.
• “There’s always enough random success to justify almost anything to someone who wants to believe.”
• “There is a strong general tendency to filter out the bad and the failed and to focus on the good and the successful.”
• “Along almost any dimension one cares to choose, the average value of a large collection of measurements is about the same as the average value of a small collection, whereas the extreme value of a large collection is considerably more extreme than that of a small collection.”
  • Example: the average rainfall for an area over a 5-year period is close to that over a 1-year period but not likely to be near the extreme over the past 50 years.
• “Because people usually focus upon winners and extremes whether they be in sports, the arts, or the sciences, there’s always a tendency to denigrate today’s sports figures, artists, and scientists by comparing them with extraordinary cases.”
• “Coincidences or extreme values catch the eye, but average or ‘expected’ values are generally more informative.”
• Expected Value = average of the possible values weighted by each value’s probability. Best used to figure out the value of gambling games.
• “Rarity by itself shouldn’t necessarily be evidence of anything. When one is dealt a bridge hand of thirteen cards, the probability of being dealt that particular hand is less than one in 600 billion. Still, it would be absurd for someone to be dealt a hand, examine it carefully, calculate that the probability of getting it is less than one in 600 billion, and then conclude that he must not have been dealt that very hand because it is so very improbable. In some contexts, improbabilities are to be expected. Every bridge hand is quite improbable. Likewise with poker hands or lottery tickets.”
• Gambler’s Fallacy = the mistaken belief that past events somehow impact the outcome of future events.
  • Ex: Because a coin came up heads five flips in a row, tails is more likely on the next flip (It’s not more likely, tails is just as likely as the last five flips — 50% likely).
  • “In terms of ratios, coins behave nicely: the ratio of heads to tails gets closer to 1 as the number of flips grows. In terms of absolute numbers, coins behave badly: the difference between the number of heads and the number of tails tends to get bigger as we continue to flip the coin, and the changes in lead from head to tail or vice versa tend to become increasingly rare.”
• “Most people don’t realize that random events generally can seem quite ordered.”
• We often try to imbue meaning to randomness, after the fact, with a neat and tidy story to make it all make sense. It’s not true, of course. You see this done daily to explain why the stock market and individual stocks moved up or down.
  • “Commentators always have a familiar cast of characters to which they can point to explain any rally or any decline. There’s always profit-taking or the federal deficit or something or other to account for a bearish turn, and improved corporate earnings or interest rates or whatever to account for a bullish one. Almost never does a commentator say that the market’s activity for the day or even the week was largely a result of random fluctuations.”
• Random streaks like sequences of heads or tails in a coin flipping contest or made baskets in a basketball game, can be predicted to an extent.
  • The odds of making 6 or more baskets in a row (assuming a 50% shooting percentage) is about 10%.
• Poisson Probability Distribution = used to find the probability of a set number of events occurring in a specific time interval. Can be used to predict rare events if you know just how rare the event is.
• “Inspect every piece of pseudoscience and you will find a security blanket, a thumb to suck, a skirt to hold. What have we to offer in exchange? Uncertainty! Insecurity!” — Isaac Asimov
• “If one’s model or one’s data are no good, the conclusions that follow won’t be either. Applying old mathematics, in fact, is often more difficult than discovering new mathematics. Any bit of nonsense can be computerized—astrology, biorhythms, the I Ching—but that doesn’t make the nonsense any more valid.”
• Because basic math deals in certainties it’s often used to make impressive claims about nonsense. Burden of proof is on those who make such claims.
• Jeane Dixon Effect = A few correct predictions get promoted and remembered while forgetting the large number of incorrect predictions. Also known as confirmation bias.
• Nobody provides an end-of the-year list of incorrect predictions by oracles, psychics, or market gurus.
• Why Believe Astrology, Psychics, etc.?
  • We see what we want to see in the vague pronouncements, imbuing them with “truth” that never existed.
  • We remember the “true” predictions, overweight coincidences, and ignore the rest.
  • We give it importance due to its age, simplicity, or complexity.
• “Medicine is a fertile area for pseudoscientific claims for a simple reason. Most diseases or conditions (a) improve by themselves; (b) are self-limiting; or (c) even if fatal, seldom follow a strictly downward spiral. In each case, intervention, no matter how worthless, can appear to be quite efficacious.”
• “Experience never forces one to reject any particular belief.”
• Conditional Probability = the probability of an event occurring given that another event already occurred.
  • Confusing the probability of A given B and the probability of B given A leads to innumeracy.
  • Used when counting cards in blackjack — the probability changes based on which cards have already been played.
  • Bayes Theorem = used conditional probabilities to figure out the probability of causes relative to an effect.
• “Disproving a claim that something exists is often quite difficult, and this difficulty is often mistaken for evidence that the claim is true.”
• The psychological impact of innumeracy leads to biased judgement. Having a good filter, a sense for numbers, and an appreciation for probabilities is one way to recognize anomalies and coincidences for what they are and not give them significant meaning.
• “Remember that rarity in itself leads to publicity, making rare events appear commonplace.”
• “The tendency to attribute meaning to phenomena governed only by chance is ubiquitous.”
• Regression to the Mean = the tendency where an extreme value of a random variable is likely followed by a value closer to the variable’s average.
• “If we have no direct evidence or theoretical support for a story, we find that detail and vividness vary inversely with likelihood; the more vivid details there are to a story, the less likely the story is to be true.”
• Framing
  • Choose between gaining $30,000 or an 80% chance of winning $40,000 and 20% of winning nothing?
  • Choose between losing $30,000 or an 80% chance of losing $40,000 and 20% of losing nothing?
  • Most people take the $30,000 in the first question even though the expected gain on 80% of $40,000 is higher ($40,000 x 0.8 = $32,000).
  • Most people will take the 80% chance of losing $40,000 even though the expected loss is higher.
  • Kahneman and Tversky prosed these questions and concluded “that people tend to avoid risk when seeking gains, but choose risk to avoid losses.”
  • We respond differently to questions and statements based on how they are framed.
• “Mathematics can help determine the consequences of our assumptions and values, but we, not some mathematical divinity, are the origin of these assumptions and values.”
• Probability Oddity
  • Nontransitive Dice
    • There are four dice: A, B, C, and D
    • A has 4 on four sides and 0 on the other two sides.
    • B has 3 on all six sides.
    • C has 2 on four sides and 6 on the other two sides.
    • D has 5 on three sides and 1 on three sides.
    • If dice A is rolled against B, or B against C, or C against D, or D against A, the former dice will win 2/3rds of the time. So A beats B which beats C which beats D beats A 2/3rds of the time.
    • Ed Thorp tells a story of Buffett playing a game with him with these dice.
    • The trick to let your opponent pick one dice first, so you can pick the dice that has a two-thirds chance of beating it.
  • Condorcet’s Paradox
    • Marguis de Condorcet proposed that majority rule was self-contradictory.
    • There are times in social decision-making i.e. voting where individual preferences lead to the majority rejecting every possible choice — where voters will prefer candidate A to B, B to C, and C to A.
• Acting in your self-interest may not always serve your best interest.
  • See the Prisoner’s Dilemma.
  • Business transactions fall into this too. Choosing a win-win scenario where both parties profit a little, rather than only you profit a lot (screw over the other party), leads to the perception that you’re fair and others want to do business with you.
• Statistical Errors
  • Type I Error – when a true hypothesis is rejected.
  • Type II Error – when a false hypothesis is accepted.
  • Type I and II Errors offer a basic mental model for weighing trade-offs.
    • Ex: FDA drug approval process. Okay a bad drug (Type I Error) or not okay a good drug (Type II Error).
    • Ex: Quality or Price.
    • Ex: Pascal’s Wager.
    • “The point is that all kinds of decisions can be cast into this framework and call for the informal evaluation of probabilities. There is no such thing as a free lunch, and even if there were, there’d be no guarantee against indigestion.”
• Survey Errors
  • Tiny sample sizes.
  • Sample size misrepresents the population — not diverse or varied enough.
  • Self-Selected Bias – Common issue in voter polling. Often a byproduct of who responds. Who answers the phone? Who’s at home? Who refuses to respond? The same is true for reader surveys – automatically self-selects for readers, leaving representative demographic of non-readers out.
  • No Confidence Interval – provides a range of value based on specified confidence level. Often presented as 95% confidence interval represented by plus or minus X%.
    • “For samples of a given size, the narrower the confidence interval—that is, the more precise the estimate—the less confident we can be of it. Conversely, the wider the confidence interval—that is, the less precise the estimate—the more confident we can be of it. Of course, if we increase the size of the sample, we can both narrow our interval and increase our confidence that it contains the population percentage (or whatever the characteristic or parameter is).”
    • “If margins of error aren’t given, a good rule of thumb is that a random sample of one thousand or more gives an interval sufficiently narrow for most purposes, while a random sample of one hundred or less gives too wide a margin for most purposes.”
• “Especially for the innumerate, a few vivid predictions or coincidences often carry more weight than much more conclusive but less striking statistical evidence.”
• Law of Large Numbers –
  • The difference between the probability of an event and the frequency of its occurrence approaches zero as the number of trials grows.
  • Said another way, the average of some measurements should approach its true value as the number increases.
  • “Succinctly: The law of large numbers gives a theoretical basis for the natural idea that a theoretical probability is some kind of guide to the real world, to what actually happens.”
• Central Limit Theorem
  • “The central limit theorem states that under a wide variety of circumstances this will always be the case—averages and sums of nonnormally distributed quantities will nevertheless themselves have a normal distribution.”
  • “Succinctly: Averages (or sums) of quantities tend to follow a normal distribution even when the quantities of which they’re the average (or sum) don’t.”
• Correlation and Causation
  • Are often because neither is the cause of the other despite being correlated.
  • A third factor could influence two variables making them correlated.
  • Sometimes two things are correlated but other factors hide the correlation.
  • Accidental correlation – pure coincidence and meaningless.
• Percentages
  • A price that drops 50%, then rises 50% is not back to even. The price is down 25% from where it started.
  • “The simple expedient of always asking oneself: ‘Percentage of what?” is a good one to adopt.”
• Semi-Attached Figure – defined by Darrell Huff as a number, taken out of context, with no information how it was arrived at or what it means.
• “When statistics are presented so nakedly, without any information on sample size and composition, methodological protocols and definitions, confidence intervals, significance levels, etc., about all we can do is shrug or, if sufficiently intrigued, try to determine the context on our own.”
• “Accounting is a peculiar blend of facts and arbitrary procedures which usually require decoding.”
• “Broad Base” Fallacy
  • Focus on absolute numbers to draw attention to something rather than the probabilities.
  • Also known as denominator blindness or denominator neglect.
  • Headlines like “Dow Drops 500 Points” might seem scary until you realize the Dow was at 40,000 — a 1.3% decline.
  • The denominator, the population size, provides much needed context on the absolute number given.
• Averages
  • “The urge to average can be seductive.”
  • The average of 100 can quantities ranging from 95 to 105 or 0 to 200.
  • The range of distribution, related to the average, adds important context on how meaningful the “average” is.
  • “Most quantities do not have nice bell-shaped distribution curves, and the average or mean value of these quantities is of limited importance without some measure of the variability of the distribution and an appreciation of the rough shape of the distribution curve.”
• It takes six to eight riffle shuffles of deck before its roughly in a random order.
• Statistical Significance – a result that is unlikely to have occurred by chance.
• Something can be statistically significant but have no practical significance. It might be more helpful, useful, or better but not by enough to make a difference.
• “There’s a strong human tendency to want everything, and to deny that trade-offs are usually necessary.”
• “Probability, like logic, is not just for mathematicians anymore. It permeates our lives.”
• “We sail within a vast sphere, ever drifting in uncertainty, driven from end to end.” — Pascal

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