We Suck at Math

In Liz Weston’s most recent Money Talk column, which I read in the Sunday Los Angeles Times, a reader was concerned about retirement income and spending:

In a recent column you repeat advice I have often read that withdrawing about 3% of my investment capital will reduce the chances of my running out of money in retirement. But that doesn’t make sense to me. I have been retired for over 19 years and I have sufficient data now to extrapolate that I could live for 100 more years with so meager a drawdown because, through good and bad times, my earnings after inflation and taxes always exceed 3%. If I am missing something, I must be extraordinarily lucky because it hasn’t hurt me yet, and at age 77 I think it unlikely to do so in my remaining years. Can you explain this discrepancy between my experience and the consequences of your advice?

Ms. Weston gives the appropriate answer: “Sure. You got extraordinarily lucky.” She properly distinguishes between those who retire into good markets (lucky) and those who retire into tough markets (unlucky) and correctly notes why and how the early retirement years are particularly important. While I think her implicit suggestion that “3%-to-4%” withdrawals are safe is too aggressive, her approach and her broad-brush analysis are correct.

What particularly interested me was a phrase in this sentence, which I have highlighted: “I have been retired for over 19 years and I have sufficient data now to extrapolate that I could live for 100 more years with so meager a drawdown because, through good and bad times, my earnings after inflation and taxes always exceed 3%.”

The reader doesn’t have “sufficient data” after 19 years – not even close.  We would need far more data than that to draw even tentative conclusions. We all suffer and are threatened by the same problem.  We are pretty lousy at math generally and we truly suck at probability analysis. 

As most of you know (and as my masthead proclaims), I advocate a data-driven approach to investing and retirement planning.  A major problem with this approach (as I have written before; see here and here too) is that we all have difficulties engaging with issues mathematically and probabilistically. 

We are all prone to innumeracy, which is “the mathematical counterpart of illiteracy,” according to Douglas Hofstadter. It describes “a person’s inability to make sense of the numbers that run their lives.” Although Hofstadter coined the term, mathematician John Allen Paulos popularized the concept with his book, Innumeracy: Mathematical Illiteracy and Its Consequences.  While illiteracy strikes mostly the uneducated, we are all prone to innumeracy. 

For example, most people would consider it an unlikely coincidence if any two people would share the same birthday in a room with 23 people in it. People would generally look at it like this:  since one would need 366 people (in a non-leap year) in a room to be certain of finding two people with the same birthday, then it seems to make sense that there is only a 6.28% chance of that happening with only 23 people in a room (23 divided by 366). However, 99% probability is actually reached with just 57 people in a room and 50% probability exists with only 23 people (see more on the “birthday problem” here).

In the investment world, we intuitively tend to think that if we start with $1,000 and suffer a 50 percent loss on Day 1 but make 50% back on Day 2 (day-to-day volatility being exceptionally high, donchaknow), we’re back to even. However, were that to happen, our $1,000 would be reduced to a mere $750 (more on the “arithmetic of loss” here).  Similarly, a sum of money growing at 8 percent simple interest for ten years is the same as 6 percent (6.054 percent to be exact) compounded over that same period. Most of us have trouble thinking in those terms.

These examples are pretty (pardon the pun) simple.  When things get more complicated we can really go off the rails, especially when the answer seems straightforward.  To illustrate, if you have two children and one of them is a boy born on a Tuesday, what is the probability you have two boys?  If you do not answer 13/27 or 0.481 — as opposed to the intuitive 1/2 – you’re wrong (to find out why go here).

The inherent biases we suffer (as discovered by behavioral finance) make matters worse.  For example, we’re all prone to the gambler’s fallacy – we tend to think that randomness is somehow self-correcting (the idea that if a fair coin is fairly tossed 9 times in a row and it comes up heads each time, tails is more likely on the tenth toss). However, as the commercials take pains to point out, past performance is not indicative of future results. On the tenth toss, the probability remains 50 percent.

We also tend to suffer from availability bias and thus value our anecdotal experience over more comprehensive data (see the cartoon below, from XKCD).  For example, the fact that most of your friends use MySpace is not enough evidence to conclude that it’s a good product (much less a good investment).

The conjunction fallacy is another common problem whereby we see the conjunction of two events as being more likely than either of the events individually.  Consider the following typical example. A group of people was asked if it was more probable that Linda was a bank teller or a bank teller active in the feminist movement from the following data points:   “Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.” Fully 85 percent of respondents chose the latter, even though the probability of two things happening together can never be greater than that of the events occurring individually.

Now suppose that Company X has a workforce that is only 20 percent female.  The base-rate fallacy would suggest that the company is discriminatory.  But further analysis is required.  If the applicant pool was only 10 percent female, Company X might actually have an exemplary record of hiring women. If you want to learn more in this area, you might start with this paper on teaching statistics.

In his 1974 Cal Tech commencement address, the great physicist Richard Feynman talked about the scientific method as the best means to achieve progress.  Even so, notice what he emphasizes:  “The first principle is that you must not fool yourself–and you are the easiest person to fool.”  The examples above make Feynman’s point. It’s easy to fool ourselves, especially when we want to be fooled – we all really like to be right and have a vested interest in our supposed rightness.  If we are going to be data-driven (that that’s a very good thing), we need to check our work and our biases very carefully anyway, and especially because we suck at math and are even worse at probability.


43 thoughts on “We Suck at Math

  1. A week ago, this article would have inspired a vigorous argument from me (I do math for a living), but then I read an article where someone asked the following: “If you were buying a baseball and a bat and the total amount you spent was $1.10 and the ball was $1.00 more than the bat, how much was the bat?” If you said $.10, you were wrong…and right…that we do suck at math. 🙂

  2. I’m a little irritated after reading your post; I kept getting the math problems wrong. I have always considered myself a somewhat smartish person yet Feynman’s point regarding how easy it is to fool ourselves is reinforced by my inability to even get half of the questions correct.

    Like most people I’m not a financial guru, just a business guy with a family that needs to make the correct decisions at least 51% of the time. In life I constantly find myself having to make decisions based on limited information with limited time to verify that limited information. In this post I can click on a link to find the “right” answer, but honestly I never even thought I needed to question that the possibility of having a second boy was anything other than 50% based on the first boy being born on a Tuesday.

    This may be the reason most of us follow the other lemmings off of the cliff, it isn’t that we can’t figure out the right answers (I believe most people are at least somewhat smart) it is that we don’t always know the right questions in the first place.

    Always great to connect with a fellow blogger, keep the posts coming!

    • Having a (new) second child is still a 50-50 proposition. This Tuesday child problem deals only with when the two kids are *already* born. If you learn that a couple has two kids and they have only a boy in tow when you meet them, you might assume that there’s a 50% chance the other kid is a boy. But you’d have better odds of being right if you guessed the other one is a girl. However, if you learned that the boy is either the oldest or the youngest, then the odds go back to 50-50 on the other child. Will this be useful in real life? Highly unlikely except as (very boring) cocktail party chatter.

      But if you rely on fallacies in your retirement planning or investing, because you don’t know the math, it can have bad consequences on your plans — and I think that is Bob’s point.


  3. Thanks for the post, Robert. Embarrassingly, I fell into many of the innumeracy traps laid out up there.

    Nonetheless, I think there are a few things worth pointing out:

    – Many of these math paradoxes are intentionally made confusing by ambiguous wording. Take for example the Monty Hall problem, or the Tuesday Child puzzle used in your post. Both rely on a vaguely worded statistical problem (“one is a boy born on a Tuesday”) which does not specify the exact parameter required to achieve the intended answer (“the statement actually rules out the other child being a Tuesday boy”). While there is a valuable lesson to be learnt there about discovering the exact meaning of a sentence, the problem there is one of literacy rather than numeracy.

    – Time spent near a casino pit will show you: it’s scary how prevalent the gambler’s fallacy is. But not everything in life follows a static, defined, roulette-style distribution function. Consequently – as someone familiar with buy-side finance – much time is spent chasing after institutions, managers, strategies which have good track records. Not necessarily because past performance is indicative of future success, but because a small subset of them do have underlying performance distributions which exceed the mean. Their past performance records are one of the tools used to assess this. [The big caveat is that doing this is, for most, a losing venture]

    – The base-rate fallacy (there’s an extra letter “L” in the link, btw) I feel may be more clearly explained here: (http://www.fallacyfiles.org/baserate.html), because in the Schneider link he seems to conclude that any system which cannot give a >50% hit rate on terrorists is useless. That is patently untrue: according to his math, NSA screening increases the probability of detection by some 4000x above the base rate which would be achieved by a coin (1.32% vs. 0.00033%). Again, there’s a good discussion to be had there (“is this sort of civil liberty tradeoff worth it?”), but to go straight to “1.32% is less than 50%, hence useless” is a bit extreme.

    I did enjoy the post a great deal though, and would highly recommend that readers take a look at the ever popular Daniel Kahneman for some fantastic explanations of some of the fallacies mentioned here.


  4. Bob — You get a +1 (times 3) on this one for working in Feynman, Hofstadter and XKCD in one post. If you’re ever in Minneapolis, I’d love to grab a beer with you.

    (a) On investors being bad at math, the easiest misapprehension is the arithmetic of loss, which totally applies to those investors who chase around mutual funds based on last year’s (quarter’s) performance. It gets harder to comprehend from there. The issue of withdrawal rates and being lucky on timing of retirement is one that has gotten a lot of attention lately. (Perhaps the best advice is to study hard, earn a lot, have low overhead, save a lot, invest wisely and take advantage of tax benefits, and accumulate enough so that the math hopefully becomes irrelevant to your retirement).

    (b) On psychological biases, I like Old School Value’s short summary here If you want to read more, Charlie Munger has a lengthy speech on inherent biases, which is reproduced in Poor Charlie’s Almanack.

    (c) As far as the Tuesday child problem, I believe it suffers from imprecision in language. Does “one child is a boy on Tuesday” mean “(exactly) one child is a boy on Tuesday” or “(at least) one child is a boy on Tuesday”? It makes a material difference, and I think the problem solver missed the boat on that one. The Monte Carlo simulation is also entirely unnecessary and shows off when the probability of any child being male (or female) is 0.5 — you can just generate the permutations, eliminate the permutations that don’t fit the problem, and calculate the fraction If you really want to code, use it to generate the permutations. I think you can probably do it with a one-line list comprehension in Python, although a few like breaks would be more Pythonic.

    (d) The Tuesday child problem also calls to mind the Monty Hall problem, which has also applicability (and more alpha) on the Deal or No Deal game show. It’s very hard for people to wrap their mind around that one, and it may require Monte Carlo simulation for those who say “show me”.

    (e) I take issue with the assertion that we are all prone to innumeracy, but not illiteracy, at times. The bar for literacy is so low compared to math. Even in English, with all of its bizarre grammatical rules, the problem is so easy compared to math. You have 26 letters, and maybe 100 or so sounds they can make. Our brains seem to be good at blocking letter patterns so we can read. Math is such a broad and deep field. If you don’t make its study your life’s work, it is difficult to master even a corner of this domain. If the bar for literacy were higher, let’s say the ability to read and write fluently in four or five languages, to discuss the merits of Infinite Jest or Porttair of an Artist as a Young Man on demand, to crank out sonnets or symbolically-rich novellas or expository writing at will, then we might not be so literate. In fact, if the bar for literacy were the ability to compose a thought and relay it plainly and concisely, very few would be considered literate, including most lawyers.

    (f) The larger problem is America sucking at math (I like the term “innumeracy”). Why is it OK for little Johnny to suck at math, but not at English? Parents who were taught (or learned) math badly dismiss the importance of math. Many people harbor the fantasy of seeing their algebra teacher 20 years hence and taunting them by saying they made it half their adult life without someone asking them to calculate where the train from Topeka will pass the train from El Paso.

    But that tries to excuse the problem we are facing — math is becoming more central to competing in the global economy. As science and technology build on the backs of past giants, the need for math/science/technology and coding skills is creeping into so many job categories. For example, even courtroom lawyers are needing to have (or hire) these skills as the business of preparing for disputes and trials becomes one of using large databases to store and sort out relevant information.

    Our competitiveness as a nation depends on parents, schools and policy-makers realizing that we need to prioritize math learning to at least the same priority level we give to youth sports.

    Anyway, best post yet!

    • Great points, all. I have written about the Monty Hall problem, by the way. It’s in the links, I think. Thanks for posting even though you have already won (Week 1).

  5. The SAT had my all time favorite question, and I think it is more “fair” than having exactly one boy born after flipping a coin 8 times and switching doors to get the goat:

    If you drove from city A to city B at 80mph in one hour, and back at 40mph what was your average speed for the trip. If you are saying 60mph you clearly haven’t been reading the previous posts. It should have taken you twice as long to drive back at 40mph… lowering your average speed.

  6. You article has also illustrated a common issue in dealing with math. Word problems suck at describing math equations. We use math equations to clearly define what results we are trying to reach. Word problems often use tricks and distractions. The problem’s writers believe that the tricks distracts people from reaching an obviously clear result. I disagree. Most people give the answer to what the believe that the writer intended to describe. People tend to describe math problems poorly and their intended results are frequently different than described. We give the “obvious” result because if it was really difficult they would then clarify their real intentions. These are the patterns that we socially learn from dealing with children, the elderly, the poorly educated, and foreigners who may have difficulty describing things well.
    You are correct that a lot of people do not understand how to formulate a problem from real life situations. If you can not change a situation into a real mathematical equation then you can not get the correct result. Many of the common issues with word problems are more due to just badly written problems that the writer believe describes a clear result. I can just look at my 4th grader’s homework to see how badly the problems are phrased.

  7. Oh wow, the more everyone comments, the more complicated this all gets and really gets mind boggling. It’s a good thing that there are knowledgable advisors out there
    (Just a side note- the probability of having two boys all depends on timing – just ask me, I have three)

  8. Karen has it right! Most people would have given up a third of the way thru your post! Only reason I didn’t is because I like the competition.
    Most people give up on managing their finances as well, The myriad of opportunities and advisors touting different choices is overwhelming to most.
    Be professional, fact find properly, have the client’s needs be primary and do your homework. Seems to work better than dazzling them with math knowledge.
    But I do love your posts!!

    • That’s great advice. The KISS method is crucial in many situation. The key, I think, is relayed by Oliver Wendall Holmes, Jr.: “I would not give a fig for the simplicity this side of complexity, but I would give my life for the simplicity on the other side of complexity.” When we take the complex and make it understandable and relatable to clients, everyone wins. Thanks for commenting.

  9. While talking to my wife about a certain government grant to pay down student loans she says, “Well its really more than $50,000 because it saves us the total amount we would have paid for the life of the loan” (i.e. principal plus interest over the payment period).

    “False” I reply, “NPV of those payments might actually be exactly $50,000 today”

    Even if you know math economics makes it all more complicated.

  10. I find it an unfortunate irony that in the investment and retirement space in particular, we are so focused on being data driven with our decisions, even though the reality we see is both that most clients don’t know how to effectively interpret or apply the data, through some combination of innumeracy, ‘behavioral finance’ biases and heuristics, or both.

    It’s a difficult challenge as a practitioner. Certainly, I don’t want to step away from the rigor of, say, a quality Monte Carlo analysis to examine a problem (assuming I’ve got appropriate parameters for it), or some other data-driven methodology of analysis. Yet it doesn’t change the fact that intuitively, our brains simply DO NOT really know how to process a decision like “is a 95% probability of success acceptable for your plan?” In the end, we humans only know how to really envision binary outcomes – it happened, or it didn’t – and we judge accordingly, then some of the best analysis is probabilistic.

    Where does the practitioner draw the line between the rigor of an analysis, and the potential disconnect it creates for the end client who must use the results to come up with a decision? Is it entirely just a matter of “good translation” – figuring out how to communicate the data-driven analysis and results in a manner that promotes good decision-making? Or do we have some greater, even more deeply rooted challenge in trying to figure out how to help a non-data-driven species to make data-driven decisions?

  11. Yes, we do suck at math, especially in probability. Much of that comes from relying on flawed intuition. But equally to blame are the educators who rely on misunderstood examples from the past, and so explain it wrong. And one of your examples, the variations of the Two Child Problem, is the poster child (pun intended) for the issue.

    If you know that a couple has two children, and that one is a boy, the probability that both are boys depends on how you learned that one was a boy. If it is because you asked them “is either one a boy?” and they said “yes,” then the chances are 1/3. But if it is because they “have only a boy in tow when you meet them,” as sdarji suggests, the answer is 1/2. Why the difference? One out of four couples will have two boys, and three out of four will answer your question in the affirmative, making the chances 1/3 in the scenario with the question. But only one in two such couples, who are towing only one child of two, will be towing a boy/ That makes the chances 1/2 in the “meeting” scenario. The difference is due to the possibility, which most people ignore, that you would learn about a girl when there was a boy you could have learned of.

    This is the exact same reason people miss the Monty Hall Problem: they dismiss just the cases where the car is behind the door Monty opens, when they need to also dismiss the cases where it held a goat but Monty Hall would have opened another door. In both problems you get the wrong answer by considering all of the cases where the information you have is true, and the right answer by considering just the cases where it is true AND you would learn that information.

    The reason it sounds strange to change the answer when “born on Tuesday” is added, is because it is very unintuitive to think you learned the fact by asking a couple “Do you have a boy born on a Tuesday?” But that is what the 13/27 answer implicitly assumes.

    And finally, it is quite ironic that you mention John Allen Paulos’ book in the same article as the Tuesday Boy Problem. Because he started a trail of Innumeracy that may have led to that problem. Except for the first two, any direct connection between these documented events is speculation on my part, but they seem plausible:

    1) In that book, Paulos asked the two-child problem about girls, and named the girl you knew about Myrtle. He said the answer was 1/3, which is incorrect for several reasons.

    2) In an article titled “Three bewitching Paradoxes,” J. L. Snell (Dartmouth) and R. Vanderbei (Princeton) noted that, by introducing the name “Myrtle,” Paulos had changed the problem. 1/3 could not be correct for the same reasons “Tuesday” would change the answer to 13/27. It should be something a little less than 1/2, depending on how rare the name “Myrtle” is. They were still implicitly assuming you learned the fact by asking specifically about girls named Myrtle, and ignored another problem entirely.

    3) In 2008, Leonard Mlodinow (Cal Tech) wrote a book called “The Drunkard’s Walk: How Randomness Rules Our Lives” where he changed the name to Florida, and gave the less-than-1/2 answer. But he acknowledged the other problem: to get this answer, you need to assume two children in the same family can share a name. He said the assumption had negligible effect; but he was wrong. The correct answer, if you assume you asked about Florida, is *greater* than 1/2 by much more than Mlodinow thinks it is less than 1/2.

    4) In 2010, Gary Foshee introduced the Tuesday Boy problem at a festival honoring Martin Gardner. I speculate that this follows from the name versions, because it removes the duplicate-name issue. Following Gardner’s original solution of 1/3 to the simpler problem, Foshee answered 13/27. Which is ironic, since Gardner retracted the 1/3 answer six months after publishing it. Not a fine way to honor him, by forgetting what he said.

    But to this day, most people, like sdarji, either have not heard of Gardner’s retraction, or refuse to believe it.

  12. But since the girl’s name was “Florida,” and not “George Foreman” (the only known example of duplicate names), we can indeed conclude that duplicate names do not occur in her family. 🙂

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