Thursday, April 26, 2007


Visualizing large numbers

Yesterday, Maggie posted about the difficulty we have in visualizing large numbers. It's a good post, and it reminds me of the many attempts I've seen to do this sort of thing.

There's a display outside the National Air and Space Museum, in Washington, DC, that tries to help us visualize the distances within the solar system: it places the sun near 4th St SW, at the eastern end of the Air and Space Museum. It then puts the planets at proportional distances, so that Pluto (yes, it's still there, as a “dwarf planet”) is over by “the Castle”. It's cute, but, while it shows the relative distances, I think it doesn't really convey the absolute distances.

In general, mapping large distances to smaller ones doesn't work very well, because we don't seem to be good at scaling things in our minds. I'm sure we've all seen the comparisons to the Earth, or areas of it: “If all the [things] were laid end to end at the equator, they would go around the Earth 17 times.” Or “If we loaded all the [things] onto a freight train, the train would have to be as long as the Mississippi River. Cool. What do those comparisons really mean? Do they really help visualize how many [things] there are?

We've often tried to use time, too, as Maggie does. “If you read one book every hour, 24 hours a day, you'd have to live to be 296 years old to read every book ever written.”[1] Or some such. “If you could count one atom every second, it would take 2.3 billion years to count all the atoms in the solar system.”[1]

Maggie's proposing another comparison with time, using the “if you did one thing every second” model. I think it's a good idea, and I think her approach to tying it into a school term is an excellent addition that might help make it real. Ultimately, though, I think it'll still fail to convey the numbers involved, for two reasons. One is that we just can't get our heads around numbers that large, no matter how they're mapped. I think we compress time in a similar way to how we compress quantities — we really don't have a good conception of the relative difference between one second and five and a half years, because the difference is too vast.

The other reason is that Maggie's addressing the odds of winning the lottery, so we're superimposing a load of other things on top of the diffculty with large numbers. We also have to include the general lack of understanding of statistics and probabilities, and the level of denial that any of that applies to me when I buy my lottery ticket. Here's a (by no means exhaustive) list of problems we'll encounter in trying to talk to the average Joe about why he's not going to win the lottery:

  1. The small up-front cost works in the lottery's favour. If it costs one dollar to buy a ticket, the expense seems so small that it's “worth a try”. At 100 dollars a ticket, there'd be far fewer takers. Of course, if you buy one ticket every week for two years, you'll spend 100 dollars... and your expected return will still be so close to zero that there's really little difference. In fact, and perhaps ironically, you'd probably have better odds of winning if you bought one ticket for $100 than if you bought 100 tickets for $1 each, because so many fewer people would play. But this point is lost on the majority of people, who just think that $1 is worth the risk.
  2. People don't understand the concept of independent events, or they forget what they might know about it when it comes to playing the lottery. Someone who's dutifully bought a ticket every week for two years often has some sense that he's “due to win”, that it's “my turn”. And if he misses a week because he's away on vacation, he might actually rue that, thinking that that was the week he would have won, had he only bought a ticket. Yes, when you look at it from the beginning, your odds of a win are better if you plan to play 100 times than if you plan to play once. But if you've already lost 99 times, your odds of a win on your 100th play are exactly the same as those on your first: vanishingly small.
  3. People, even when they ought to know otherwise, have a sense that they can “get close”, and that it matters. I heard someone say this last week, in fact: “Oh, man, this is the closest I've gotten! I bet I can get it next week.” No, sorry: if you have to match six numbers to win, and you matched five of six this time... that gives you no edge whatever toward matching six next time. Next time, and every time, you start from scratch.
  4. People maintain an unreasonable level of optimism with respect to these sorts of things. “Someone has to win. It could be me!” Well, yes, it could be. It could also be you who gets squashed by a boulder falling from the hill you're driving past. And you know what?: The latter is the more likely one. But we tend to be optimistic about favourable events with minuscule probabilities, while at the same time thinking that unlikely disasters with greater, but still small, probabilities are simply too remote to consider.
  5. People believe in “luck”. This is sort of related to the optimism thing, but rather than just reflecting an attitude, the “luck” question is just one of magical thinking. “I feel lucky today.” I think I can beat the house, but just today, just because I felt good when I awoke, or because I found a shiny penny, or because the stars are right. Even if one can convince someone of all the points above, one can't compete with magical thinking.
I actually have discussed all of these with lottery-ticket buyers, so I'm not just speculating here.

Maggie will surely help some of her students understand the astronomical probabilities involved in the lottery. I'll be interested in hearing how it goes with the class, and how they respond to it.

In the end, though, they're still going to buy their lottery tickets — and some may actually do it just to try to prove that the teacher is wrong!

[1] No, I made those up; they're just examples of the sort of thing I'm talking about. I've actually seen comparisons like these, but I have no recollection at all of the numbers they gave. Which kind of demonstrates my point, actually.


Maggie said...

It is an inspiring problem, and perhaps a losing battle. Perhaps the wrong battle. Maybe we shouldn't try to get people to "understand" large numbers. Perhaps we should be addressing the emotional basis of the misunderstanding.

For example, having read Robert Cialdini's Influence (and I'm sure I reference it ad nauseum), I try to label instances of influential phenomenon when I encounter them in life. For example, reciprocation. An obvious example is the "free sample." You feel inclined to reciprocate the favor by buying the product. But if you recognize that it is a manipulative technique, you can either forgo the sample in order not to feel the urge to reciprocate, or enjoy the sample knowing it's part of the advertising cost of the product.

Labeling and recognizing the phenomena doesn't make them go away, but I think it might make me less susceptible.

Perhaps if we teach people to recognize their emotional responses to number/probability phenomena, we can innoculate them against the effects.

Or maybe a two-pronged approach, in which we work on both aspects at the same time. Practice would be essential.

But I'm not teaching any subject directly related to everyday probability, effective newspaper reading, or number sense, I'm teaching two programming courses. So at best, it'll be a little gedankenexperiment that I'll carry through the semester. I can't just be experimenting on my students! (I do justify the idea by saying that computer technology deals in very large numbers, but that's a bit of a stretch. I would feel it fit in with the computer fluency curriculum, but I'm not teaching fluency next semester.) I will informally publish my informal results on my blog.

If you have any further ideas on the subject, please share them. My daughter might be doing a science/math education thing for one of her Girl Scout service projects (silver or gold), and I think a lottery module would be interesting. It's something I've wanted to do, so even if she doesn't end up choosing it as her project, I might continue with it myself.

scouter573 said...

Part of the trick of the Lottery is that there's no perceived penalty. I'm restating Barry's first point, but I think you'll agree that losing $100 at one time is "more painful" than losing $1 this time (and 99 more distributed across time). Our Infinite Capacity for Rationalization kicks in, too.

The other calculation I did once was to calculate how much it would cost to buy enough tickets to give me a 50% chance of winning. Or to give me a 100% chance of winning. If you really believe in a lottery, mortgage the house and buy the tickets. (Well, a BIG house...)

Alex said...

Both this and Maggie's post were interesting. I think it's important not to restrict this to just the lottery example. I get far more frustrated with people's inability to perceive and properly use large numbers when discussing the daily news. Two points about the lottery.

* Less people playing does not increase your chances of winning. It increases your chance of not having to share a jackpot. A single $100 ticket does nothing for you (though with far fewer people playing the jackpot would probably end up being less.) 100 $1 tickets at once will increase your odds by a factor of 100. (If we wanted to mess with people's minds in the opposite - evil - direction, we could tell them it's a 10,000% improvement in odds!) Playing Powerball, your chances of winning the jackpot go from 1 in 150 million to 1 in 1.5 million. In other words from "practically impossible" to "absurdly unlikely." However, you will likely win back from $3 to $10, and have improved chances of the other prizes. (see the odds)

* Many if not most people who play the lottery see it not as an investment but as a game. The only return they expect is the excitement of the drawing. Lots of people waste money on various forms of recreation. Most of them offer no chance of becoming a multimillionaire.

Barry Leiba said...

Alex is right, of course. When I said, "In fact, and perhaps ironically, you'd probably have better odds of winning if you bought one ticket for $100 than if you bought 100 tickets for $1 each, because so many fewer people would play," I was thinking of a drawing, where one ticket is drawn from the set of tickets that were purchased, rather than of the "match the numbers" sort of lottery, where the total number of tickets is not relevant to whether your numbers match.