Logical Risks

January 27, 2006 12:02 PM

Answer three simple tricky questions and predict your sensitivity to risk. via Washington Monthly
posted by alms (73 comments total)

I liked those questions. (Getting them right made me feel smart.)
posted by oddman at 12:20 PM on January 27, 2006

Those questions are a joke. Princeton students had an average score of 1.63?? Wow.
posted by Gyan at 12:23 PM on January 27, 2006

It's astounding that Princeton students, on average, only got half the questions right.
posted by monju_bosatsu at 12:25 PM on January 27, 2006

I got all three questions right, but I am a woman. This makes me not so risky, right? For me, at least, that's spot on.
posted by arcticwoman at 12:31 PM on January 27, 2006

I got them right. But I really had to force myself even to answer them.

Internal monologue: Of course I'm a risk taker. Why do I have to answer a bunch of stupid math questions to find that out?

So, I am not actually patient, just stubborn.
posted by Miko at 12:34 PM on January 27, 2006

Three out of three, and I hate math.
posted by Faint of Butt at 12:36 PM on January 27, 2006

3/3, haven't taken a math class since junior year of high school.
posted by scody at 12:37 PM on January 27, 2006

Maybe, suggested Professor Frederick in the interview, "it's because one study was done at University of Toledo" — where the mean score on his test was 0.57 out of a possible 3 — "and one study was done at Princeton," where the mean was 1.63. The test groups may not really be the same.

Gee, ya think?
posted by brain_drain at 12:37 PM on January 27, 2006

I was also surprised at how low the scores were. I think this is a sign of people trying to elide the math and just game the question based on what it sounds like the answer should be.

That said, I was even more astonished (apparently) the majority of people would take a sure $500 over a 15% chance at $1,000,000. That just doesn't make any sense to me at all, unless maybe you are talking to someone who has a long history of living hand-to-mouth, and/or someone who has basically never had anything break their way in the past.

"For instance, 80 percent of high-scoring men would pick a 15 percent chance of $1 million over a sure $500, compared with only 38 percent of high-scoring women, 40 percent of low-scoring men and 25 percent of low-scoring women."

Is there anyone here who would take the $500?
posted by alms at 12:42 PM on January 27, 2006

I always bungle those "5 machines, 5 widgets, 5 minutes" type of questions. Every time. I found the others extremely easy, and do very well with math problems, but something about that problem always gets me, probably because I'm confident in my cursory assessment and don't double check (hence the "risk" assessment of the test). I always look back on it and say "Oh, of course!", but blow it the first time through.

So don't be so quick to call out Princeton students for getting half of them right on average. The questions are designed to specifically test where your latent problem solving skills fail, and as such I would expect the average score to be average. If the average was a score of 3, that wouldn't be a very well designed test, would it?
posted by team lowkey at 12:43 PM on January 27, 2006

I don't think I'd take the $500, but i wouldn't pay $500 for a 15% at a mil
posted by edgeways at 12:49 PM on January 27, 2006

OK edge, would you rather pay $500 cash or have a 15% chance of incurring a $1 million debt?
posted by alms at 12:53 PM on January 27, 2006

alms, there have been times in my life that I would have, no question. Not now, no, but, oh yes, there have been times.
posted by MrMoonPie at 12:53 PM on January 27, 2006

The logic that a 15% chance at 1 million dollars is worth $150,000 is based on the assumption that you get an infinite, or at least statistically significant number of tries at the game.

If it's a one time thing, a sure thing might be the way to go.

I mean, which would you rather have? $500 or an 10^ -19 chance at 1 * 10 ^ 22? If it's a one time only offer.
posted by Capn at 1:07 PM on January 27, 2006

Shouldn't we have to qualify the results of this test by saying "All natural risk (immediate risk of starvation, death or injury from exposure, etc etc) being eliminated, this is how what we found about risk tolerance"? I mean, if you asked the question "Would you rather have $500 now or a 15% chance at a million" of someone living on the street, I wouldn't qualify a preference for the first option as risk avoidance, I'd qualify it as survival. All of these college students are basically answering these questions from the perspective of someone not in want.
posted by spicynuts at 1:08 PM on January 27, 2006

Is there anyone here who would take the $500?

I definitely would. And the majority of high-scoring women agree -- in fact, the majority of three subgroups agree.
posted by booksandlibretti at 1:12 PM on January 27, 2006

True...it depends on what $500 means to you. What if the figures were changed to a guaranteed $1 million or a 50% chance at $10 million? I'd wager that many more people would take the sure thing.
posted by rocket88 at 1:13 PM on January 27, 2006

rocket88 has it. I'd much rather have $500 than a pie-in-the-sky chance of a mil.

I'm also a hardcore pessimist though, so I tend to equate a 15% chance to nothing.
posted by Firas at 1:29 PM on January 27, 2006

Got 2/3, by the way, the last one totally threw me. My first instinct was to look for a root or fourth-root of 48, but obviously that wasn't going to be a whole number...
posted by Firas at 1:43 PM on January 27, 2006

I got 3/3, am very risk-averse, could really use $500 right now, but would jump at the 15% chance at a million. That's better than one in seven.
posted by solid-one-love at 1:50 PM on January 27, 2006

3/3, suck it haters.
posted by OmieWise at 1:57 PM on January 27, 2006

I wouldn't take the $500, but if it got much closer to the value of the 15%, I'd probably take a lower payout. $149,000? Definitely. $125,000? Definitely. Much lower than that and I'd have to think about it.
posted by lackutrol at 2:01 PM on January 27, 2006

Okay, goddamn it, I'll ask:

How does $1 + $0.05 = $1.10? Sales tax??
posted by LordSludge at 2:03 PM on January 27, 2006

Oh, and got 3/3 and am a guy. I get to see weird approaches to risk every week at my poker game.
posted by lackutrol at 2:04 PM on January 27, 2006

LordSludge, it's that $1.05 + $0.05=$1.10. A dollar is not a dollar more than $0.10, it's $0.90 more.
posted by lackutrol at 2:05 PM on January 27, 2006

Oh, fuck me...
posted by LordSludge at 2:07 PM on January 27, 2006

LordSludge, x + (x + $1) = $1.10, hence x = $0.5.

They're all trick questions though, I'd be surprised to see them on an SAT test. Or maybe I remember the SAT wrongly, but I'm pretty sure it's most of a test of 'how fast can you solve this' rather than 'are you really solving this the right way? are you sure? are you triple sure?'
posted by Firas at 2:11 PM on January 27, 2006

Yeah, I probably would have missed at least two of these if I hadn't been thinking "now, why is the obvious answer wrong here?"
posted by lackutrol at 2:14 PM on January 27, 2006

Yes, looking at the low average scores reported in the article, you have to wonder how the questions were presented. If someone just handed me the questions on a piece of paper and gave me a pencil, I probably would have gotten some wrong. But since they were presented as trick questions, I knew to be extra-special crafty in taking time with them.

Maybe that's where the correlation to patience and risk-taking comes in?
posted by alms at 2:19 PM on January 27, 2006

I'm with lordsludge: the ball question boinked me [do'h!], the others were fine.

I think it's less about math and more about 'now, really, think this one through'. In fact, they're test of perception, forethought and reasoning, not numeric ability.

I'm sure if you stopped me on the bus I'd have said 10 cents, 100 minutes and somewhere around 24 days, without even thinking about it.
posted by jrochest at 2:39 PM on January 27, 2006

They are testS of forethought...

think before hitting post button, think before hitting post button....
posted by jrochest at 2:40 PM on January 27, 2006

I got the three questions easily. I wonder though if they were mixed in with other more traditional word problems whether I would have performed as well. I probably would have to be honest but it would have been more difficult because I would have had to ascertained on the fly that the questions were "tricky" instead of determining this from the context as a group of question or maybe even from the article.
posted by I Foody at 2:44 PM on January 27, 2006

I agree with jrochest and Firas: these questions really test your capability to analyze and reason, not mathematics skills. It's very important in today's world to read the meaning and not just isolated factlets that jump out at you.
I'd guess it's because formulating things in a misleading way that does not actually state a wrong implication is becoming more and more common:
Think of politicians and their varied use of statistics: when is it the number of crimes that's going up, when is it the crime rate?
If you're looking for a credit card, does it offer "free" anything, and is it really cheaper to get one month free, eleven months at $6.99 than one whole year of monthly payments of $6.40?
I would guess many people are duped using psychological tricks like those mentioned in the article, and those doing the duping are in the clear because, technically (legally), they've made no false claim.
posted by PontifexPrimus at 3:15 PM on January 27, 2006

I would guess many people are duped using psychological tricks like those mentioned in the article, and those doing the duping are in the clear because, technically (legally), they've made no false claim.

PP, I think you've just summarized the basis of our entire consumer economy.
posted by flug at 3:47 PM on January 27, 2006

3/3. I thought the lily pad question was the most "intuitively obvious". My first thought was right shift!, then ethernet!. The widget question seemed the least intuitive. I had to think of it in terms of a single robot's production capacity. I didn't immediately recognize that both situations had a 1:1 robot/widget ratio.

Coincidentally, I've been thinking about risk quite a bit lately. I haven't quite determined my personal risk tolerance, at least in regard to money. During the past couple years I've been involved in a fairly risky financial situation. The kind of "all in" gamble only suitable for single people in their 20s. Starting out, I was worried I wouldn't be able to quit while ahead. But in the end, I was actually quite relieved to get out. Even though the scheme was incredibly successful, I feel no desire to try for more. It's as if the earlier gamble has made me risk-averse, even though I won. That seems counterintuitive.
posted by ryanrs at 3:49 PM on January 27, 2006

Two things:

1) Of course these psychological tricks work. People always make fun of infomercials selling things at $19.99 rather than $20, but they do it because it works. We have an emotional barrier at certain price points and a lot of people don't think about it very much. Same with "three free months of cable" or "sample issue of our magazine." This sort of thing appeals both to a need for instant gratification and to several different kinds of laziness.

2) It's very difficult to answer a question like "$500 or a 15% chance at a million" without knowing the marginal utility (I think I'm using this term correctly) of the $500. If you had a gun to my head and were demanding $500, I'd certainly take that option. I think "risk takers" in this context means both a) people who understand the value of the "risky"option and b) people for whom the $500 is less (marginally?) valuable.
posted by lackutrol at 4:00 PM on January 27, 2006

Weird, I found the widget question utterly obvious, but couldn't understand the ball question until I came here. Still only 99% sure I get it.

I suck at math but am a voracious casino gambler, for what it's worth.
posted by CunningLinguist at 4:02 PM on January 27, 2006

CunningLinquist: Makes perfect sense.

alms: I'd take the $500 over an 85% chance of nothing. I can make quick use of $500, but nothing I already have. I wonder how that question correlated to present personal debt?
posted by ?! at 4:17 PM on January 27, 2006

lackutrol, I think that might explain my decreasing risk tolerance. As the returns accumulated, the potential downside became greater while the marginal utility of future returns decreased. Also: rocket88.
posted by ryanrs at 4:24 PM on January 27, 2006

?!: I'd take the $500 over an 85% chance of nothing.

What happens the other 15% of the time? Your proposal "feels" different because it is incomplete.
posted by ryanrs at 4:30 PM on January 27, 2006

3/3, although I was a longtime subscriber to Games and, to be fair, I had to suppress the urge to "immediately" answer with what I instinctively sensed was a false answer. The phrasing is very, very tempting (as is the desire to skim the article down to where the answers are written). This speaks to our natural love of shortcuts. It's also an excellent warning about how many polls are structured.

lackutrol is absolutely correct about marginal utility. If I had a guaranteed annual income of >$75K, as I did a couple of years ago, I'd happily take the gamble for the higher amount. Right now, though, I'd grimace and pocket the $500 cash, thinking all the while of whether to spend it on shoes, RAM, or food. I've experienced both states and I know very well how it affects my decision-making.

Now, I'd like to see this thought experiment extended to Presidential elections ...
posted by dhartung at 4:33 PM on January 27, 2006

If it's a one time offer, i'd definitely take $500 over $1m. It's crazy talk not to.

As far as the questions went, the first two took me a moment to spot the trick (but it was obvious there was one) and the third was easy. But then, as a computing student, i'm used to thinking in powers of two...
posted by iso_bars at 4:34 PM on January 27, 2006

I think the question has less to do with risk, and more to do with how you spend money. Like, if I got $500, I'd probably put it in the bank and spend it slowly on slightly better things than I normally would buy for myself. I would hardly notice I had it, and if pressed for a detailed account of how I had spent that $500, I wouldn't be able to tell you. Winning $500 is something that won't affect my life in any real way. It's a 100% chance of getting something unmemorable. If I take the $500, I'm always going to be wondering if I would have gotten the million. If I don't get the million, well hell, it was only $500.
posted by 23skidoo at 4:45 PM on January 27, 2006

iso_bars, as a computing student, would you give up your computer for a 1 in 7 chance of $1m? Maybe couch surf for a month?
posted by ryanrs at 4:47 PM on January 27, 2006

I would take a 15% chance at $1 billion rather than a sure $500,000. Raise the stakes 10x and I think I'd take the $5 million.
posted by ryanrs at 4:53 PM on January 27, 2006

Powerball Lottery Prizes and Odds
posted by ryanrs at 4:57 PM on January 27, 2006

What happens the other 15% of the time?

The other 15% of the time you get screwed out of it, anyway.
posted by dirigibleman at 6:59 PM on January 27, 2006

Would you rather take $5 or have a 15% chance at winning $10000?
posted by bikerdriver at 7:21 PM on January 27, 2006

If it's a one time offer, i'd definitely take $500 over $1m. It's crazy talk not to.

Only if you would seriously harm yourself by not having an extra $500, or if you would gain nothing from $1mil.

And there are plenty of people who would have to look long and fucking hard at that $500 before turning it down, but there are, I think, also a lot of people who would very much like a quick $500 but don't need it badly enough to sacrifice a chance at such great goddam odds.

Part of the problem is the absurdity of the proposal. If someone walks up to you on the street and somehow credibly offers you this choice, you have to be in dire straits to turn down the gamble, because it's $500 you didn't have anyway. Whereas if someone was asking you to gamble your $500 paycheck, money you expected to have, then, well, shit. That's a sudden considerable economic speedbump for many, many folks.
posted by cortex at 7:25 PM on January 27, 2006

Would you rather take $5 or have a 15% chance at winning $10000?

See, that's the thing: it's a question of the actual impact of the windfall, not just the relative proportions.

Anybody, seriously anybody would be fucking nuts not to take the 10K. $5 is not enough to tempt -- if you're in things so goddam bad that $5 looks good to you, you've probably learned pretty well how to do without $5; on the other hand, if you're anything but very wealthy, $10K is a serious situation-changer. (I could cut my college debt more than in half! And that's a very low-impact thing; what if I had serious medical bills?)

So compare that with $500 vs. $1M. $500 could be more of a windfall -- you could have incurred a desperate one-time expense that needs mitigating immediately, for example -- but $1M would fundamentally change most people's lives in a way that the $10K above would not. (I could pay off my college loans, buy a house, quit my job, and live off the interest of the remainder as a bohemian, unsuccessful artist!)

It's only when the sure thing begins to be a life-changer that it makes a good bet. For people who are not utterly desperate, the self-control involved in making the calculated risk (minority but significant chance at fundamentally changing your situation for the better) is the only way to go.

But some people are desperate, and still more are probably short-sighted enough that they'll decide they're better off with the guaranteed payoff than a proportionally ridiculously generous chance at striking oil.
posted by cortex at 7:33 PM on January 27, 2006

If someone walks up to you on the street and somehow credibly offers you this choice, you have to be in dire straits to turn down the gamble, because it's $500 you didn't have anyway. Whereas if someone was asking you to gamble your $500 paycheck, money you expected to have, then, well, shit. That's a sudden considerable economic speedbump for many, many folks.

What's interesting is that, from an economic analysis standpoint, in both situations you've paid $500 for the chance at a million. I wonder how economists would incorporate what you pointed out into the difference in choices (ie., $500 you expected to have vs. $500 that you'd get as bonus, but immediately). A sort of reverse marginal utility? Marginal hurt, heh.

It's only when the sure thing begins to be a life-changer that it makes a good bet. For people who are not utterly desperate, the self-control involved in making the calculated risk (minority but significant chance at fundamentally changing your situation for the better) is the only way to go.

I don't think that's quite correct. An extra $500 won't change my life, but wow, it'd make my next month more pleasant. I'd still go for for the $500 though. The general principle you're underscoring is correct, but in this circumstance would only work if such things were continuous—ie., if you got a similar choice all the time and turned it down. In that sense the question is faulty because it's trying to extrapolate from one decision to every decision you make.

What I'm saying is that, the very chance of being given a choice between a certain $500 vs 1/7th chance of $1,000,000 is so low that it colours the outcome.

(Sidetrack: Another interesting thing would be to correlate answers of this 'guaranteed high payout vs. probable high payout' to 'gauranteed low payout now vs. gauranteed high payout in installments for a decade'.

Specifically, I wonder if people with bad spending habits would go with a low payout immediately despite knowing that an installment based payout would be much better for them in a behaviour sense and also end up giving them more money in a 'just the numbers' sense.)
posted by Firas at 8:01 PM on January 27, 2006

There are also psychological factors at work, given that we're led to believe that anything that sounds too good to be true probably is; anything offering a million dollars 'just for the heck of it' is very very unlikely to end up providing you with a million. Else you'd be hitting every 'punch the monkey' ad you see...
posted by Firas at 8:03 PM on January 27, 2006

Granted, but that's a fundamental problem with the question. Anything offering $500 'just for the heck of it' -- especially a guaranteed $500, no question asked -- is also ridiculous. Not to say that you aren't right to suggest some psychological bias, but the question should be treated as a credible offer, on both sides, or it is meaningless.

Testing the credulity with which a variety of folks approach a variety of offers -- that is, how they would judge such offers, how much credit they would give them -- would be an interesting seperate topic, regardless.
posted by cortex at 8:25 PM on January 27, 2006

I thought the lily pad question was the most "intuitively obvious"....The widget question seemed the least intuitive.
yep, same for me.
I am also surprised so many people would take the $500. I'm a pretty broke student, so at first I was thinking it was surprising there would be people who were even more desperate than me, enough to feel like $500 is significant enough to give up a decent shot at life-changing income. But then it occurred to me that perhaps it's the reverse - maybe people who would take $500 are basically financially comfortable, have what they need, and basically just have less interest in "life-changing" money because they're pretty happy with life as it is, and some extra cash would be fun, end of story. For me, $500 would not have much impact - I would still be in lots of debt, and wouldn't necessarily feel like I had permission to go crazy with it - I am just in a phase of life where everything is on hold when it comes to spending money.
posted by mdn at 8:32 PM on January 27, 2006

3/3.

I would always take the bigger gamble. If someone is offering me guaranteed cash or a chance at a much greater amount of money, by going for the big payoff, even if I lose, I'm not walking away with anything less than what I came with.
posted by Meredith at 8:35 PM on January 27, 2006

A guaranteed $1 million is an interesting price point. I could retire someplace cheap or keep working in San Francisco. Versus a 15% chance for $2 billion. I think this would be a Hard Decision.
posted by ryanrs at 8:44 PM on January 27, 2006

Furthermore, I believe reducing the payout ratios to $1 mil vs. $100 mil does not make the decision any easier.
posted by ryanrs at 8:53 PM on January 27, 2006

Now that I've been thinking about the issue awhile and am not just defending my initial choice, I'll agree with cortex; unless it'll make a big difference for a long term, going for the certain lower payout is probably the worse idea.
posted by Firas at 8:56 PM on January 27, 2006

Money means so many different things to people that measuring their risk preferences through the filter of their desire/need for wealth seems a bit limited. Sure, it's interesting if you are only interested in the risk preferences when money is on the table but it would seem to preclude wider conclusions about risk taking.

If the quality at stake was something other than money then you probably could make wider assumptions about the impact on risk preference of gender and intellectual ability. However, the only alternative quality I can think of which is a universally appreciated value, would be the length of your life. But it would be a bit difficult to get respondents to take the debt risk preference question seriously...

And my first answers would not all have been correct if I'd been casually asked them, rather than warned they might be tricky. I'd probably have got the widgets one wrong simply because I'm lazy/hasty. The other two though seemed completely obvious (sorry LordSludge). Maybe because I've irritated myself too many times doing the instinctive thing in a problem solving puzzle that I always resort to thinking in formulae.

Thanks for the links and discussion.
posted by pots at 4:06 AM on January 28, 2006

I'd still pay a dime for that ball.
posted by Balisong at 9:06 AM on January 28, 2006

Hee hee, I got 3 out 3 and I actually go to the University of Toledo. I didn't take the test on campus though.

And even though I'm a starving student, I would take the 15% chance at one million without hesitating.

I play a lot of dice-based games, and so I feel as though I have a pretty good idea of what a 15% chance means. I would guess, that to a lot of people, 15% just means "low" and might mean the same thing whether it was any percentile from 5-24%. Maybe that's what the test is measuring.
posted by benimoto at 9:09 AM on January 28, 2006

If it's a one time offer, i'd definitely take $500 over $1m.

I'd take the chance at the million. I already have $500, and $500 more won't appreciably change my circumstances. It might make the next month or two easier to get through -- but I can get through them fine without it. But a million? Invest that money for a decade or so and I could retire twenty years early. A 15% chance at being able to retire before the age of fifty, at no net cost to me? Yeah, I'll go for that; I'm feeling lucky, and besides, I'm half-Irish.
posted by kindall at 9:20 AM on January 28, 2006

And I've always heard that the lottery is a tax for people who can't do math.

Guess it's the other way around.
posted by squarehead at 12:27 PM on January 28, 2006

squarehead, if the lottery offered these odds, it'd cease to exist after the first massively disportionate payout to those of us who can do math.

Everybody knows the house comes out ahead; otherwise there wouldn't be a house anymore.
posted by cortex at 4:18 PM on January 28, 2006

cortex, I was being glib. But one of the basic points of the article was that those who completed the (math) problems correctly (particularly males who completed the problems correctly) were more likely to take financial risks for potential profits (even when the expected value of the riskier outcome is less than the expected value of the less risky outcome). I quote:

Getting the math problems right predicts nothing
about most tastes... But high scorers — those who
get all the questions right — do prefer taking risks.

Even when it actually hurts you on average to take
the gamble, the smart people, the high-scoring
people, actually like it more," Professor Frederick said
in an interview. Almost a third of high scorers
preferred a 1 percent chance of $5,000 to a sure $60.

So I think my point stands. The lottery has been ridiculed by some as a tax for people who can't do math, whereas this article seems to imply the contrary. I realize, of course, that I am ignoring the sorts of considerations mentioned above of different economic situations entering into consideration of weighting of expected outcomes, hence my own characterization of my comment as being glib.
posted by squarehead at 6:09 PM on January 28, 2006

Fair enough. The $5K vs. $60 example is much clearer to your point. And I agree that the "tax on the stupid" characterization of major lotteries is missing in part the whole allure of megabucks gambles -- it's not that people expect that they're getting fair odds on millionairedom, it's that they like the fantasy that they might in fact beat the odds with some lightning-strike luck.

What I said in an earlier comment about nominal-vs-lifechanging prizes should clarify my position on the $60 vs. $5K question, though: if $5K will have a measurable impact on my life that $60 will not, I might be in smart to take the slighty poor odds for the bigger prize, insofar as I can spare an almost $60 and could make very good use of $5K. And that, I suppose, is your point, and the point of the cited paragraph.

However, that presumes that I am basing it on a one-time trial, and that I am following my gut in that situation instead of my math. Mathematically, yes, I'd be stupid to take the chance at $5K, and I'd be particularly stupid to take it for repeated trials. And so, yes, the lottery argument so defined.
posted by cortex at 6:33 PM on January 28, 2006

3/3. I don't entirely understand the people that wouldn't take a 15% chance at the million.
posted by The Monkey at 7:24 PM on January 28, 2006

3/3, but I wonder how I would have done on those questions if they had been among 97 other, less tricky, questions, and it was 12:30pm and I was a hungry college student and I'd get to eat lunch with my friends as soon as I completed the 100 goofy questions.
posted by Holden at 12:28 AM on January 29, 2006

Almost a third of high scorers preferred a 1 percent chance of $5,000 to a sure $60.

see, this doesn't really make sense to me... I think whether or not you would be willing to put your own money in is worth asking. Would you pay $60 for a 1/100 chance to win $5000? I wouldn't. But would you pay $500 for a 1/7th chance to win $1million? I would.

But of course, no house would ever offer such odds, because then you would just come up with $3500 to turn into a sure million bucks. The reason lotto is for dummies is, as above, that the odds are never in your favor. If you can do the math, you can work out whether or not it is actually sensible to take the risk. Because $3,500< $1,000,000 and $6,000>$5,000 it is clear which of these is a "smart" risk and which isn't.
posted by mdn at 11:57 AM on January 29, 2006

But of course, no house would ever offer such odds, because then you would just come up with $3500 to turn into a sure million bucks.

But not quite, mdm. $3500 would buy you seven independent trials (presuming this is not a raffle of tickets-without-replacement), which comes out to not a sure thing but merely reasonably good odds:

For one trial, p = 1 - (6/7) ~ 15%.

For seven trials, p = 1 - (6/7)^7 ~ 66%.

So you've still got about a one-in-three chance of blowing your $3500.

Similarly, you've got a one in nine chance of blowing twice that; at the $10K mark, you're still running at only 95% chance of winning.

Of course, that depends on the lottery system; my numbers assume identical independent trials, as I said. In a real lottery, if there would be a fixed number of valid combinations, and if you're suggesting that this is a one-ball lotto with only seven values -- number 1-7 on seven balls in the hopper, one of which will be chose -- then, yes, you could buy seven different tickets and be sure. (And I recall a story [possibly apocryphal] of a group of investors gaming a lottery at one point by some such trick.)

But (and I can't believe I'm saying this in this context), that would be an absurd proposition. Much more appealing is the notion of the eccentric billionaire offering not a numbered ticket but rather a coinflip, essentially.

Besides which, the better-or-worse odds in a true lottery are mitigated by the fact that, with better odds of winning, more winners are likely to turn up, and the house will undoubtedly split the winnings evenly rather than paying out the full sum to each. So to take advantage of some unlikely instance of house-unfriendly odds, you have to not only recognize the odds and take advantage of them but also expect that no one else (or at least sufficiently few other parties) will notice the same thing and capitalize similarly.
posted by cortex at 12:36 PM on January 29, 2006

I also answered 3 of 3. I took the $500 question this way:

Crazy TV show calls me to the stage. Hands me $500 and says "Walk away with $500 or reach into this pit and pull out one of the million dollar balls. There are 150 winners in that pit of 1000 balls. If you don't win you give us back the $500 and leave."

If I lose you and I know I am walking off the stage no worse off than when I walked into the studio. However, I am walking off the stage $500 poorer than when I stepped on stage. And at home I have to answer why I gambled away two months of groceries.

If I didn't need the $500 I would take a chance. Why not?

Cortex: Please correct me if I'm wrong. It has been many years since I took math, but in my scenerio don't the odds stay at 15% no matter how many times I try? (Assuming I toss the losing ball back in each time.)
posted by ?! at 8:17 AM on January 30, 2006

Yes, ?!, they do stay the same each time. And that's why 7 tries is not a guaranteed winner.

Recast it in slightly more intuitive terms: a coin flip.

Let's say heads you win a million bucks, tails you win nothing.

Flip the coin once, you've got 50% odds. (More formally, probability of winning p = 1 - (1/2) = .5 )

Flip the coin twice. Do you have 100% odds, now? Nope. Two independent 50% trials. You have, in fact, 75% chance of winning. (p = 1 - (1/2)^2 = 1 - .25 = .75 ) The math is easy: you're multiplying your odds of losing together by the number of chances you have. So take the chance of losing, and raise it to the power of the number of trials, and subtract that from 1, and there are your odds.

And so it goes. Each successive independent trial -- coin flips, 1000-ball-pool-with-replacement pulls -- improves your odds without making them decisive. With coinflips, your chance of losing is cut in half with each trial (since you win as soon as a heads comes up, and it's twice as unlikely to show three tails in a row than two tails in a row, and so on for n+1...). For the 15% chance, the odds of losing decrease more slowly.

In any case, the odds get very, very good over time, but it's an asymptotal graph -- you get close, but you don't get there, unless we start talking about limits and infinity.

Paging Zeno!
posted by cortex at 8:29 AM on January 30, 2006

yeah, it's true that it makes a difference whether it's a 'closed' system lottery, so to speak, where you could actually buy every option, or whether it's what I would think of as an open system kind, where every chance is still a chance, and theoretically you could flip tails every time. I would definitely be tempted to try to invest enough to win that money if it were a flexible system (ie, I could team up with friends and invest x in order to make our odds of winning the million high enough - but you're right, no matter how many tickets we buy, if it is really a coin flip (with a seven sided coin :)) we still have a chance of losing every time.)

?!'s interpretation of the risk is more honestly in keeping with the question's original intent, of course - it's a one shot deal that you can't game. Still, my instinct is to go for it just because the odds are reasonable - I probably won't win, but I realistically could win - it would just be everyday good luck, not insane, unfathomable good luck... In fact the insane good luck could be said to be having the chance to have such a decent shot at it!
posted by mdn at 10:10 AM on January 30, 2006