On April 17 1986, a young woman presented herself at Heathrow's gate 23 for that morning's El Al flight to Tel Aviv. She had cleared the airport's own security checkin procedures, but to El Al's security staff something didn't appear right. A search of her hand luggage revealed 1½ ounces of Semtex and a detonator, hidden in a calculator.Of course, I'm starting to give the impression that we should worry about terrorism, when in fact doing so has surpassed the war on drugs as one of the worst acts of collective hysteria since McCarthyism. For example, think of dying prematurely as an enormous waste of time (like reading blogs! but worse.) The average adult has about 300,000 hours of waking life to expect, and a 1 in 13,000,000 chance of dying on the average U.S. domestic flight (this was true in 19901999 entirely from accidents, and in 2000present almost entirely from 9/11, as there's been only one fatal accident during that time. Your chance of dying on a developing world flight is 1 in 1.5 million, regardless of carrier. These and other fascinating facts found here). If you're riskneutral, avoiding this risk should be worth two minutes of your time. By this argument, two minutes is also a good baseline for the amount of time that airport security should be willing to waste per passenger. The last point is the crudest part of the argument and shouldn't be stretched too far, since it's based on the risks we currently see, which might change in the extreme case of airports completely eliminating screening. Needless to say, I think we're far from that point. (Incidentally, google for probability terrorism dying to see how a little math knowledge can be an upsetting thing.)The young woman was Anne Murphy, a white, Catholic girl from Dublin. The explosives had been planted by her boyfriend, Nezar Hindawi, a terrorist with links to the Syrian government.
The time vs. death comparison seems a generally useful one because it avoids all the problems in comparing lives with money. Another place it's obviously relevant is in speeding. The probability of dying while driving is supposedly proportional to the fourth power of your speed. In the U.S., drivers die at a rate of about 15 per billion miles and are seriously injured at about 10 times that rate. If we assume the fourthpower law and that everyone is driving 65mph (assume a spherical cow...), then increasing speed by 1mph will save 0.84 seconds/mile and increase the chance of being killed/mile by about 9.2 x 10^{10}. Multiply by 300,000 hours and you get just about one second, which is pretty close to the amount of time saved. To account for injury, one method is to examine the number of disabilityadjusted life years (DALY's) lost. In 2002, 1.18 million died in traffic accidents and 38.4 DALY's were lost, which comes to about 33 DALY's/death. This means my "50 years/death" was an overestimate and increasing 1mph actually costs closer to 0.65 "disabilityadjusted life seconds"/mile. Even if we assume there's another car involved half the time, this is close to a tie, meaning that 65mph is close to optimal. On the other hand, this still hasn't included property damage (which ranges from 1% of GNP in developing countries to a stunning 2% in rich countries), higher gas consumption and the fact that speeding tickets waste time too. So I think this means that

we shouldn't speedModerate speeding is a little dangerous, but not unreasonably so.  Applying the 300,000 hours argument to airline security can't be totally absurd since it leads to different conclusions in different contexts.
 I should stop wasting so much time writing silly blog posts.
Update: Thanks to Aaron in the comments for pointing out mistakes in my driving calculations (now corrected); I had represented driving as 10 times more dangerous than it actually is.
7 comments:
"this was true in 19901999 entirely from accidents, and in 2000present, entirely from 9/11 (!) as there have been no fatal accidents during that time ..."
Commuter crash, August 27
Good point. The source I used is dated Jan 19, 2006. Since only 246 of the 9/11 fatalities were on the four planes, this means the 20006 risk is 20% higher than I had said. On the other hand, adding nine months to the denominator reduces the risk/flight. I get an adjusted result of 2 minutes and 8 seconds, unless I'm missing other crashes this year.
Another way to describe this risk, by the way, is that if each flight lasts three hours, then you'd have to fly 24 hours/day for 3000 years before you have a 50% chance of dying on a flight.
One more about profiling from my new favorite blog.
Interestingly, now when you google probability terrorism dying your blog comes up #3. Now you know how to get to the first page on Google.
So we have:
D(v) = # deaths/mile = A * v^4
Let K = Integral(p(v) v^4 dv)
Current death rate R = Integral(p(v) D(v) dv) = Integral(p(v) A v^4 dv) = A K = 1.5 * 10^(8) for current distribution over v.
T = Wasted hours per deaths.
Hours used per mile = 1/V
Total time per mile used is AT v^4 + 1/V.
Optimizing gives 4AT v^3  1/v^2 = 0, or v^5 = 1/(4 A T).
Using your velocity distribution of delta spike at 65, A = R / K = 8.4 * 10^(16), and T = 300000.
This gives V = 63 MPH as least amount of time wasted. This isn't terribly far off from the 65 assumed.
Yes, this ignores accidents, speeding tickets, etc.
I would argue that A should be lower, as (a) It's not a spike. The v^4 weights higher speeds higher by quite a bit in the v^4 portion of the integral (replacing the delta by a gaussian of mean m and width w gives K = 3w^4 + 6w^2m^2 + m^4, rather than m^4), and (b) The p(v) is in terms of velocities weighted by miles travelled, not by cars on the road. Among other things, miles are traveled faster at higher speeds, so more miles are travelled by those going faster.
I'll also argue that we shouldn't optimize time, but rather current value of time. By the same argument that money in the future is not worth as much as money available now, I'll argue that time now is also worth more than time later. So we should replace T = 300000 with a discounted version. The easiest thing to do would be to assume a straight exponential decay of worth. Even a very small discount rate (a fraction of interest rates, say) can heavily discount future rates,
These both end up raising the optimal speed a bit. (Try 75 + 5, which is what I see on my highways)
Thanks Aaron! Doing those multiplications on a napkin caused me to miss a factor of ten (now corrected). I've corrected the post accordingly to show your point that 65mph is close to optimal. Also I think DALY's use timediscounting, which is why the number of DALY's lost is only 33 times the number of deaths (from traffic accidents). On the other hand, if we think the future will be better than the present, then we'd like to spend more time there, and maybe should even have negative timediscounting.
I'm not sure I agree that A would be lower if we correctly modelled the actual distribution of speeds. (a) According to your formula, changing a deltafunction on 65mph to a Gaussian with width of 10mph would only increase K by 14%, and a width of 15mph would increase K by 33%. (b) This assumes that everyone is always driving on the highway. The EPA estimates that Americans drive 45% of their miles on highways and 55% in cities, which would cut K by more than 50% if the average speed in cities is half the highway speed.
One other point I came across in the WHO report is that the rate of injury/accident scales as V^2, serious injury/accident as V^3 and death/accident as V^4. But we also need to know the rate of accidents/mile as a function of speed. Our calculations all assume this is constant; on the other, I think the results aren't too sensitive to this assumption.
You're quite welcome.
I'm certainly biased to driving being on highways, because I live close enough to work to walk, so driving is often to, say, Santa Fe, rather than a daily commute.
Yes, the DALYs do seem to be discounted  but 300000 waking hours seems to be ~ 50 years average, which seems a bit high if it's discounted.
Yes, I had been assuming the v^4 was per accident, rather than per mile.
One sentence leapt out at me from http://www.dft.gov.uk/stellent/groups/dft_roads/documents/page/dft_roads_506880.pdf
> In the more generalised form (Baruya et al., 1999), this research identifies that the change in accident frequency for each 1km/h reduction in mean speed is inversely related to the current mean speed.
However this isn't clear to me whether this is per carmile, per cartime, per roadmile, or what.
I think this can change the v^4 anywhere from v^4 to v^6, but I'm not quite sure.
Anyways, interesting problem.
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