Quantitative Illiteracy is driving me absolutely batshit

The BBQ Pit
41

But they both increase utility

Quite.

Most people’s utility curves are somewhat “S”-shaped. As such, they are happy to either:

  • Sacrifice a little money to ensure they don’t lose a lot of money
  • Sacrifice a little money to have the chance of winning a lot of money.

Although in one case uncertainty is decreased and in the other it is increased, they are both cut of the same cloth.

Or, to put it another way, the key is the semi-variance of the risk; up-side in the lottery example and down-side in insurance.

So you can see jjimm, there’s more to the choice of doing the lottery than a straight expected wealth calculation. The punters are more economically sophisticated than you think!

pan

42

Kabbes: Pembroke. Best college of them all of course. :wink:

43

Because I’m forced to, of course.

44

**
This is so true. In the most impressionable years of early education, we have generalist teachers who are expected to acquaint us with the wonders of numbers when they themselves are probably more likely to be innumerate than the average person. That’s why many of them go into primary education, which usually doesn’t demand great mathematical knowledge. [Disclaimer] This statement is based on anecdotal evidence [/disclaimer].

**
I had to take Algebra I twice, and never made it to Algebra II, let alone Trigonometry or Calculus. I did take “Calculus Lite” (aka calculus for humanities majors) in college, and did fairly well at it, but it wasn’t very rigorous. Nor could it be, because the members of the class didn’t have the necessary algebraic skill to truly understand the underpinnings of integration and differentiation.

**
For me it was discovering the principle of mathematical induction. Instead of solving a particular “problem”, you could solve an infinitude of cases! Then I proved the volumes of the sphere and the cone.

**
Trig still scares me a bit, though I have to learn it in order to proceed with further educational plans. All those ratios to remember, gah!

I find visualization very helpful as a mental exercise.

See my first statement above.

45

I take it that the percentile score was measured against the school district and not against nationwide students?

46

Percentiles on this test are measured against nationwide and statewide norms. I do realize that it would be technically, mathematically possible for a school district to reach the stated goal, but given the sample size (~30,000 students) having 90% above the 50th percentile is a practical impossibility.

47

My verbal SAT score exceeded my mathematical SAT score by a good 90 points but I can still enjoy mathematics. I got through calculus and statistics well enough to know how little I know.

Count me among those who believe that almost anyone can learn mathematics if they put their mind to it. My ex-wife was convinced that she couldn’t do higher math when she went from straight "A"s in basic mathematics to "C"s in algebra, mostly due to a teacher who didn’t take the time to teach her in language she could understand.

She just figured that as an artist, she just wasn’t mentally configured for mathematics.

When she recenly went back to school to learn and get certified as a veterinary technician, she had me tutor her in algebra and trigonometry. She ended up getting "A"s. It was mostly a matter of explaining things in more than one way, with one of them eventually sinking in, and the “aha” light coming up in her eyes.

I think that this can happen with anyone, provided they have enough incentive and desire to learn.

48

I got an 800 on the SAT math, and I had a hard time with calc in college. It is one of the major reasons I took this year off.

I agree; I didn’t really want to learn calc. It was also the first math that I didn’t instantly understand, which makes me empathize with the people who don’t instantly understand any math. The fact that my calc teacher tried to cover about three times as much as what was supposed to be in the class*, and that I never did my homework didn’t help any.

I’ve actually disliked math since early elementary school, because they made us do speed tests on multiplication. I’ve never been fast, so I always did badly, but later I found that being fast isn’t everything and is actually one of the least important parts.

I think some innumeracy problems can be attributed to poor reading comprehension. I always understood story problems, but I was constantly reading. There seemed to be a correlation between people who didn’t like to read and those who didn’t do great in math.

*By midterm, when I stopped going to class, the class was less than half as large as it had been at the beginning. The majority of people switched to other classes just to get away from the teacher.

49

Just to say, I’m also in the “Anyone can learn maths” camp. The problem is that for some people it will require a LOT of work on their part and on the teacher’s part, often more than they’re willing to put in.

50

Or put it this way, despire prior protests to the contrary, I’ve never actually found anyone to whom I couldn’t teach maths.

51

How does this utility curve make people willing to buy lottery tickets? I assume you’re talking about a curve like the one in Prospect Theory, which goes through (0,0) and is concave towards the horizontal axis on both sides of 0. The concave curve for gains should mean that a big gain with a low probability is worth less than its expected value. Unless people are neglecting the small cost of a lottery ticket as “petty cash” or greatly exaggerating the probability function, it’s hard to see how a lottery ticket could be a worthwhile monetary investment.

For most gambles, people are risk averse when it comes to gains and risk seeking when it comes to losses. So the lottery doesn’t fit in well, unless they’re fudging the numbers.

Insurance doesn’t fit into that analysis either, but that’s probably because people are losing a lot more when they need insurance. A one in a hundred chance of losing everything is worse than a 100% chance of losing one hundredth of everything. The same doesn’t hold with gains, since there’s no such thing as gaining everything. Also important with insurance is salience - more people will buy flood insurance right after a flood, even though that’s when another flood is least likely to happen. And more people will buy insurance for something exciting like a fire than for something dull like a flood, even if the probabilities & expected damages are the same. This could be part of what’s happening with the lottery - being a millionaire is a salient outcome.

You could argue that people enjoy playing the lottery, and that they’re basically just buying that experience. But, just looking at money and the utility of money, playing is irrational, since lottery winners aren’t even much happier than other people.
[/hijack]

52

78% of quoted statistics are made up on the spot :stuck_out_tongue:

53

knock knock, try the following typical utility curve:

exp(x/1E6) - 1

Lottery tickets cost $1. You have $1. You have a one in 1 million chance of winning, but win $0.5m if you do win. Your expected winnings from playing are -$0.5. Ie you expect to lose 50 cents.

Your expected utility however is 1.474E-5 if you do play and 1E-6 if you do not.

Therefore your net utility is increased from playing the lottery.

Or to put it another way, losing that dollar will make bugger all difference to your life, but winning 5 million would transform it.

Insurance works a similar way, in the opposite direction; losing something very valuable is disasterous compared to the cost of paying the insurance premium.

Suppose you have a $5.5m net worth, including something worth five million dollars. Insurance costs you $30k and there is a 0.5% chance that you lose the item.

(Certain*) wealth if you use insurance = $5.47m

Expected wealth if do not use insurance = $5.475m

Therefore taking out insurance costs an expected $5k relative to not taking it out.

However utility from taking out the insurance is 1.867, whereas expected utility from not taking it out is 1.864. Therefore expected utility is increased by taking out the insurance.

This doesn’t even allow for the fact that there is also value associated with the certainty of the insurance result and the excitement of playing the lottery.

The effect is exacerbated if (as is usually the case) the curve turns concave for higher wealth values and you are considering insurance of the substantial part of one’s wealth and lottery tickets that cost near nothing.

pan
*ignoring issues surrounding non-performance of insurance

54

I thank God for Innumerate people!!! Because they are so common, I get paid good money for a job that is really not all that hard. :slight_smile:

55

Also, is it not true that even among those who major in fields like engineering or physics, all but a very gifted few have to really work hard to master higher math? From what I’ve heard it isn’t like falling off a log even for them. It’s not like they can read a calculus book and take it all in as if they were reading a P.G. Wodehouse novel.

I think there’s a hump in math; before you get to it you’re just working on unrelated individual problems without much of a big picture. Once over the hump, you begin to consider generalizations and proveable theorems, and every equation or “trick” that you use, you should be able to derive from first principles. The average person, perceiving the perseverance that math demands, may wrongly conclude that they are just hopelessly innumerate and shouldn’t bother trying. So they never get over the hump. IANAMMajor, but that is my opinion FWIW.

56

Actually javaman, I think you just described my experience with a maths degree perfectly.

pan

57

>> You know, I read somewhere that around 50% of people are below average when it comes to statistics.
Sorry I don’t have a cite to back this up

So the median is something you see on the highway?

58

Sorry it’s been so long, kabbes, but I’ve been out of town.

Where did you get your “typical utility curve”? I don’t understand how it can be typical. All the research I’ve seen has the curve concave down, not concave up like the exponential function you give. That’s because there are decreasing marginal returns for gains, with money as with anything else. You should get less utility out of your second million dollars than out of your first. If you got more utility out of what you bought with your second million than you should’ve bought that stuff with your first million.

Looking at gambles, how many people would take a 50-50 shot at $10,000 over a guaranteed $5,000? Not many, I believe. You’d have to offer closer to $12,000 before I would take the gamble, and I think most people would say something similar. But your formula says that the $10,000 (p=.5) is preferred, with an expected utility of .005025 compared with .005013 for the $5,000.

Obviously lottery tickets are rational for some utility curves, but not for the utility curves that I’ve seen in the literature on the psychology of decision-making. I’d like to know why you think your exponential function is a typical utility curve.

59

Sorry - you’re quite right: my utility curve is rubbish. Or rather, they do tend to be convex right at the start but then they rapidly become concave.

However, they can also have discontinuities due to extreme changes of lifestyle certain wealth can offer ($X allows me to retire, but $X-1 doesn’t, possibly?). This can explain why lottery gambles can be positive.

But look at it this way: since utility curves are so difficult to pin down, we would normally reverse engineer them if we are going to work them out at all. This is done by giving someone a questionairre regarding various financial choices. On these grounds, the utility gained by playing the lottery is by definition positive!

pan

60

I think javaman hit it on the head. A lot of people never get the big picture of math, and just see it as a random collection of unrelated problems. I didn’t really get the big picture completely myself until I took a course in advanced logic (and that’s not a remedy that I’d recommend for the average innumerate), but after that, oh boy did I get it.