(This lecture is largely based on Paulos, J.A. 1988. Innumeracy: Mathematical Illiteracy and its Consequences. New York: Hill and Wang. Excellent book--buy a copy!)
Innumeracy is the mathematical equivalent of illiteracy, but is far more widespread and socially accepted than illiteracy. Innumerate people often--
Here are some problems that may help you overcome some number numbness. You're supposed to get the right order of magnitude, not the exact answer:
Irrational responses to risk: Of 28 million Americans who travelled abroad in 1985, 17 were killed by terrorists: 1 death per 1.6 million. (And 1985 was a bad year, terrorism-wise!) In contrast, of 240 million Americans who travelled on U.S. highways in 1985, 45,000 were killed in traffic accidents: 1 death per 5,300. Yet many people are more scared of terrorists than car accidents. You could probably make good (albeit somewhat dishonest) money selling nervous travelers special death-by-terrorism insurance; yet most states need laws to force drivers purchase car insurance and wear seatbelts.
Paulos estimates the risk of contracting AIDS from unprotected heterosexual intercourse with an AIDS-infected partner is 1 in 500. The chance of not contracting AIDS from a single sexual contact is 499:500. The chance of not contracting AIDS from two contacts is (499/500) squared, or from N contacts is (499/500) to the Nth power. (You could have daily unprotected sex with that person every day for half a year, with only a 30 percent probability of getting AIDS.) If you use a condom, your chance of AIDS from a single encounter with an AIDS-invected person is less than 1:5,000: you're far likelier to be killed in a car wreck. If you don't know your partner's HIV status but he/she is not a member of a known risk group, your chance of AIDS from a single sexual contact without protection is 1:5,000,000; with protection it's more like 1:50,000,000. You're likelier to be killed in a car accident driving to the drug store for condoms. Especially if you don't wear a seatbelt.
Paulos argues we should try using a logarithmic scale to understand relative risks. What is the typical American child's annual risk of kidnapping by a stranger? (Less than 1 in 5 million). What is the same child's annual risk of being killed riding a bicycle without a helmet? (Greater than 1 in 5,000). The innumerate person is unimpressed by this difference, and isn't really satisfied by anything short of absolute zero risk. Distracted by the lurid nature of kidnappings, the innumerate person dismisses the relative probabilities: "Yeah, but what if it's my kid who's kidnapped?" This kind of thinking is why we have irrational policies such as the Delaney Amendment barring food additives which can be shown to cause any additional cancers in lab animals at any dose. (Megadosing rats with distilled water causes higher incidence of tumors.)
People are overly impressed by mere coincidences. Did you know JFK's secretary was named Lincoln; and Lincoln's secretary was named Kennedy? They both had 7-letter names too. Was there some sort of cosmic force behind their assassinations? (Enquiring minds want to know!) Some "coincidences" are much more likely than you think: the chance that of 23 randomly-chosen people 2 will share the same birthday is 50 percent.
Some coincidences can be engineered not to look like coincidences. At the start of football season you could mail letters to 64,000 people predicting the outcome (by point spread) of a football game: half the letters pick one team, half the other. The next week you mail predictions to the 32,000 people who received correct picks the first week; the third week 16,000 predictions to those who received two correct picks; the fourth week 8,000 predictions to those who received three correct picks; etc. Near the end of the season you have a few dozen people absolutely convinced of your predictive powers, and you can charge them lots of money to tell them how to bet really big on the next game.
Pyramid schemes: An easier (and just as illegal) way to make money is to start a Ponzi (pyramid) scheme. Suppose 10 of you each pay me $100, and each recruit 10 of your friends to pay $100 each (which we split evenly), and get the 100 friends to each recruit 10 of their friends to pay $100 (which we split 3 ways). I collect $1,000 (10 x $100) from each of you, plus $5,000 (100 x $50) from your friends, plus $33,000 (1,000 x $33) from your friends' friends, and turn a game over to each of you. You simply collect from the next recruitment of friends-of-friends-of-friends, and for your $100 initial "investment" you get a total of $39,000 too. People only play these games if they expect to get to the top of the pyramid themselves. But each iteration of this game requires 10 times as many players as the last, so only 9 iterations would require a billion players.
People can be fooled by averages. Suppose during the first half of baseball season A bats .300 in 200 at-bats, while B bats .290 in 100 at-bats; and during the second half of the season, A bats .400 in 100 at-bats while B bats .390 in 200 at-bats. So A has the higher batting average in both halves of the season. But B has the higher batting average for the whole season: A got 60 + 40 = 100 hits in 300 total at-bats; B got 29 + 78 = 107 hits in 300 total at-bats. Why is this result so counterintuitive? Because people tend to average averages.
People look for, and perceive, patterns in randomness. Pattern-finding is the essence of learning, and reflects a natural human need to make sense of the environment. But we are often misled by specious patterns.
Innumerate people are often victims of pseudo-science. Science involves testing of refutable hypotheses (Popper): scientists don't really prove hypotheses; they can only disprove them. The current scientific paradigm is the underlying set of beliefs about our universe which have not (yet) been falsified by the accumulated body of experimental evidence. Pseudo-science tends to rely on hypotheses which are not falsifiable.
Karl Popper criticized Freudianism for its immunity to falsification. (Therapist: "You're obviously in denial about this." Patient: "No, I'm not.") Medicine is particularly prone to quackery, because most medical problems are either (a) self-correcting; (b) self-stabilizing; or (c) fatal, but exhibit uneven patterns of decline and temporary improvement. Survivors may engage in post hoc, ergo propter hoc thinking, crediting the quack treatment for their cure; non-survivors aren't around to complain.
People often misunderstand probabilities. Here's another con game: a man shows you 3 cards, one black on both sides, one red on both sides, one red on one side and black on the other. You draw one from a hat and only look at one side. (Assume it's red.) Since your card can't be the one that's black-black, he bets you even money your card is red-red. Although this sounds like a fair bet, there's really a 2/3 chance the card is red-red!
People can be misled by how questions are framed. Which of the two scenarios (taken from Kahnemann and Tversky) is more likely?
The first scenario is more likely by definition because it has one less condition. But people may think the second scenario is more likely because it conforms better to some mental model of Ann (stereotype of a feminist).
Here's another framing problem: Choose between a sure $30,000 or an 80% chance of winning $40,000 and a 20% chance of winning nothing (expected win: $32,000). Most people choose the sure $30,000. Now choose between a sure loss of $30,000 or an 80% chance of losing $40,000 and a 20% chance of losing nothing (expected loss: $32,000). Here most people take the gamble. Kahnemann and Tversky conclude people are risk-averse regarding gains, but risk-taking regarding loss avoidance. Is this win-loss asymmetry consistent with what we know about contingent valuation analyses of WTP and WTAC for environmental amenities?
A similar asymmetry is involved in hypothesis testing. Strictly speaking, a researcher either rejects a hypothesis or fails to reject it. (Hypotheses can never actually be proved, only disproved.) Researchers can be wrong in either of two ways:
The cost of being wrong may depend on what type of error occurs. Most scientists would much rather be criticized for being too skeptical (Type 1 error) than too gullible (Type 2 error).
We are awash in phony statistics. The statement that "Nine out of 10 kids prefer Skippy" may be statistically valid if a reasonably large number of kids were surveyed. It is merely an anecdote with no statistical significance at all if only 10 kids were surveyed. The advertisement should include some sort of confidence interval, but doesn't. (The confidence interval varies inversely with the square root of the sample size: here if N=100, 90% plus-or-minus about 3% prefer Skippy.) Most people don't understand statistics well enough to demand better information, and don't hold advertisers, politicians, public advocacy groups, etc. fully accountable for the bad statistics they promulgate.