Monday, May 07, 2007

LittleBig Numbers

Winding chain of thought I had: Stopped by Akusai's posts on "Warrior School" -> Native Americans/Indians/whatever -> South Park episode featuring Indian casino owners taking over South Park's land in a reversal of the Greedy Evil White Guys vs. Spiritual Environmentalist Indians theme -> Indians infected the South Parkians with SARS via blankets that they rubbed Chinese people on (Don't look at me: I don't do their writing) -> Stan heals his dad with "folk remedies" -> alternative medicine -> Earlier scene: "Stan, I only have a 98% chance of surviving."

So, 2% mortality rate. In the small scale of one person, those odds aren't bad. I laughed until I thought about the large scale. Say a disease with that mortality rate infected 1,000,000 people. That'd be 20,000 dead. I'd say that merits a fair bit of caution.

One of the things that gets repeated quite often on the medically-themed blogs I go to is that most people can't assess risk, and I believe it. Imagine that there's a 1 in 1,000 chance that when you drive your car, you'll end up in a fatal crash. Would you still drive? I wouldn't: It may take a while, but over time, those odds would add up over time. Not in the gambler's fallacy way, but the more trips I plan on, the more times I'll exposed to that small chance.

So, any other probability discussion?


Anonymous said...

Risk assessment is one of those things where context really does play a sizable role.

As an individual, a 2% mortality rate seems like great odds.
If you're a hospital serving a population of 1 million people, a 2% mortality rate means you'd better have a good turnaround time between the morgue and the mortuary.

Same story with flying on planes versus driving. Tiny risk either way, but the airlines are more likely to be sucessfully sued, so they have a fiscal incentive to lower risk. The importance of the odds of having the wrong surgery performed in a hospital depends highly on whether you're a patient (risk of a single event) or the hospital (aggregate effect on a large population)

"Not in the gambler's fallacy way, but the more trips I plan on, the more times I'll exposed to that small chance."

Ah, but this is exactly what those folks are talking about when they discuss poor risk analysis. If your initial risk assessment is wrong, that error gets massively magnified over time. Let's use the car-crash example. If the odds of a fatal accident really were 1:1000 trips, then taking two trips a day has a 50/50 mortality rate after 1 year. But why would we think the odds are that high?

A quick Google search for "automobile" and "fatality" turns up a rate of 14.66 fatalities per 100,000 resident population for 2005. We get 38,444 fatalities in 2004 out of a population of 237,961,000 registered drivers. Either way, your estimate of the odds is off by a factor of at least 10. And those are per annum numbers, which means the odds of dying sometime this year in a car crash might be 1:10,000, but your odds of dying in a car crash per drive is incredibly lower. And at 1:10,000 per year, the aggregate effect is less than 1% over your lifetime.

That's the trouble folks have: we tend to wildly over-estimate the actual risk, largely (I suspect) when the consequences are dire.

Bronze Dog said...

Guess I should state it a little more explicitly: The 1 in 1,000 was just a nice round number I threw out for the sake of an example, and I used driving because it's a behavior we engage in quite often, in an effort to demonstrate the idea that small risks can add up if you do it a lot.

It wasn't intended as an estimation of reality.

Dikkii said...

Probability and risk assessment is something bound to get me irate.

Dunno if you remember the Ford Pinto, but it had this problem where if you collided with it a certain way, it would explode.

Ford got an actuary to calculate the probability of how many Pintos were going to explode, and then, based on that, worked out how much they stood to lose through lawsuits, settlements, compensation claims, etc.

And then, this is the part that makes me sick - they subtracted this from the cost of doing a full recall and worked out that it would be cheaper to keep the crappy cars out there so that they could continue to occasionally explode.

It didn't appear to register with them that people who died due to this would be, like, you know, totally dead due to something totally avoidable which Ford execs could stop.

Not to mention injuries.

Not sure what the actual probabilities were, though.

Anonymous said...

I guess I could have made my point better.

1:1,000 is a nice round number. 1:10,000 is also nice & round. 1:100,000 is equally round.

Without breaking out a calculator, it's difficult to know how quickly or slowly those risks aggregate compared to each other. If I told you your odds of winning a lottery were 1:10,000, would you buy a ticket? What if I told you the odds were 1:100,000? Would that really, seriously change your mind about playing once, or even playing multiple times? What if your odds were 1:1 million versus 1:10 million? Still playing?

There is a threshhold of perception, and below that threshold, all risks appear equally meaningful to the casual observer. People say the numbers "stop being real" or that "It's all basically zero anyway", even though it's not. The odds of being struck by lightning are different than the odds of being bitten by a shark or carried away in a tornado, but because of the magnitude of those numbers, people tend to value them all equally.

Similarly, people might understand risk aggregating, but without doing the math to examine it, that little bit of knowledge becomes useless. The odds of winning my inter-state lottery are 1:63 million; if I play every drawing (twice a week) for 10 years, the odds of having at least one winning ticket in that 1,040 drawings has to accumulate to something useful, right? If I play every drawing for 50 years, the probabilities should aggregate into something meaningful, right? If I buy two tickets each drawing...

Of course, the aggregate odds in each of those scenarios remains well below even a 1% threshold.

That's the problem with risk assessment. Say your doctor gave you a pill to take daily. There's a 1:100,000 chance you'll die from it, but otherwise it's safe. How long would you feel safe taking the pill, knowing the risk? What if the odds were 1:1 million? How long then? What about 1:10 million?

Those odds look similar, but aggregate differently. 1:100k has a 20% mortality rate over 60 years. (or just under 22,000 pills) 1:1M stacks up less than 3% over the same time period. 1:10M is 0.2%. But to the casual gaze, one in a million isn't significantly different from one in ten million, as they're both "practically zero".

It's that perceptual problem of 'practially zero' that fouls up the average person's risk assessment.

An Anonymous Coward said...

Reminds me of this review, where the writer goes on and on about how silly it is for a character to worry about a "two percent" chance of disaster, in the process demonstrating that he has absolutely no concept of what "two percent" means:

I mean, I'm pretty sure that every time I cross a street, there's at least a two percent chance I'll get hit by a bus. I would even go so far as to guess that when I open a soda bottle, there's at least a two percent chance the plastic top will shoot off, ricochet around the room, and bury itself in my brain.

Which, of course, explains all the millions of soda-bottle-top-in-brain-related deaths every year in the U.S., given that there are no doubt many millions of soda bottles opened each year, and one in fifty times the bottle top ends up embedded in someone's brain. Yeah, that sounds reasonable.