Potato paradox
en.wikipedia.org
en.wikipedia.org
If you could chose to increase the average fuel economy of a car, which of these would save the most fuel:
1. 10mpg -> 12mpg
2. 12mpg -> 15mpg
3. 15mpg -> 20mpg
4. 20mpg -> 30mpg
5. 30mpg -> 60mpg
Of course with a little thought, it turns out that they all save the same amount of fuel, but it's hard to wrap my brain around the fuel savings being equal between 10mpg-12mpg and 30mpg-60mpg.
100 / 10 = 10
100 / 12 = 8⅓
10 - 8⅓ = 1⅔
100 / 30 = 3⅓
100 / 60 = 1⅔
3⅓ - 1⅔ = 1⅔
100 / 10 = 10 gallons used
100 / 12 = 8⅓ gallons used
10 - 8⅓ = 1⅔ gallons saved
---
100 / 30 = 3⅓ gallons used
100 / 60 = 1⅔ gallons used
3⅓ - 1⅔ = 1⅔ gallons saved
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As a side note: I worked in the factory that make Chevy Tahoes when they started making 2 mode hybrids. The hybrids didn't make more highway mpg, but they improved city milage from 16 mpg to 20 mpg. One of the problems is that doesn't sound like much of an improvement, but it's actually 1 1/4 gallons saved per 100 miles.
https://www.autotrader.com/car-news/chevy-tahoe-hybrid-great...
If I get 25 mpg and have a 12 gallon tank, that's 300 miles per tank. The nice thing about this is that it's a fixed value with simple, understandable units that more or less doesn't change as long as I keep the same car, presumably for many years. It helps me plan trips—I know that I regularly visit family who live 110 miles away, so anything with less than 220 miles to the tank forces me to refuel any time I see them… which affects my plans. America being less dense than Europe might partially explain why mpg is so popular here. A trip of 110 miles (or 200, 300) is not unusual in America. I remember making a series of trips to pick up a family member in college some 250 miles away, and that’s because she went to one of the closer options!
The same problem (averaging reciprocals) shows up in investment. If you get 10% growth this year and 10% decline next year, the average is not (1.1 + 0.9)/2 = 1.0 = 0% growth, it is 2/(1/1.1 + 1/0.9) = 0.99 = 1% decline.
Wouldn't that mean 30mpg -> 60mpg is more likely to save fuel on average because people are need less fuel to get them where they're going?
Or did I just get bamboozled by the trick?
50000 miles / 12 miles per gallon = 4166 gallons
50000 miles / 10 miles per gallon = 5000 gallons (833 gallons saved)
50000 miles / 60 miles per gallon = 833 gallons
50000 miles / 30 miles per gallon = 1666 gallons (833 gallons saved)
I'd go with choice 5 as well. Maybe it's a world where the same number of gallons is saved (per miles driven) as choice 1, but it's a world where the roads are safer.
If you are going X miles, the difference in gas usage between A mpg and B mpg is (X/A)-(X/B) = X(B-A)/(AB). So in all these pairs, the constant is (B-A)/(AB), in this case 1/60. So each adjustment means your fuel usage goes down by 1/60 of a gallon per mile.
Thus speeding by 10 miles per hour makes much more difference in your time if it happens at 20mph (3 minutes per mile -> 2 minutes per mile) than if it happens at 60mph (1 minute per mile -> 0.86 minutes per mile).
This can sometimes be erased because one typically (at least in the US) spends more time at the highway speed, but if we're talking about you need to spend 10 minutes getting on the highway, 30 minutes driving on it, and 10 minutes getting off it, then even allowing for 5 minutes of the city driving to be non-speedable (it's spent stuck at three traffic lights, say), you can save 5 minutes of time by speeding 10mph on the only 5 miles of city streets, but only 4 minutes of time by speeding 10 mph on the 30 miles of highway. You're speeding the same amount for 6 times the distance and something over twice the time, but because your average speed was higher it simply doesn't buy you as much.
Working out the numbers also helps you realize that speeding on the highway to get a substantial amount of time is, while not pointless, more unsafe than you think. Even under great circumstances, like if you are facing 40 minutes of driving -- if we're talking about a 70mph highway and you are 15 minutes late while you think 5 minutes late is still socially acceptable, you need to cut 10 minutes and thus average 4/3 * 70mph = 93.3 mph to make that happen. That means that to handle the moments where you are stuck behind two cars both going 10 over the limit at 80mph, you will need to at times be going 30 over the limit. And that's with a relatively long commute! If it's a 20 minute commute you have to drive this recklessly just to shave 5 minutes.
If you drive 50% in electric mode at 110 MPGe and 50% in gas mode at 40 MPGe, what's your average MPGe?
It's only: 1 / (0.5/110 + 0.5/40) = 58.7 MPGe
Answer: impossible. You'd have to travel the second mile at infinite speed to achieve this.
In your proposed solution, the first mile takes two minutes, and the second mile forty seconds. That's 2:40 to cover two miles, or an average speed of 45 MPH.
At an average of 90mph, the second mile took 40 seconds.
So your average speed was 2 miles / 160 seconds, or 45 mph.
(120*30 + 40*90) / (120 + 40) = 45 MPHHowever, if you've already driven the first mile at 30 mph, it's taken you two minutes to drive it. In order to then average 60 mph over the full two miles, you'd have to travel the remaining mile in 0 seconds -- in zero time.
1. 10 gal/100mi -> 8.3 gal/100mi
2. 8.3 gal/100mi -> 6.7 gal/100mi
3. 6.7 gal/100mi -> 5 gal/100mi
4. 5 gal/100mi -> 3.3 gal/100mi
5. 3.3 gal/100mi -> 1.7 gal/100mi
Of course, vehicles with better fuel efficiency tend to travel further (not a lot of tractors driving across country or whatever). So the priority should be (a) get the least efficient vehicles off the road entirely, (b) reduce total miles driven, (c) switch long distance trips away from grossly inefficient vehicles, (d) improve efficiency of other vehicles.
Increased petroleum/carbon taxes in places where people drive inefficient vehicles would be a big help (possibly offset by some kind of subsidy so the change doesn’t hammer the poor, ideally universal basic income or the like).
The overwhelming majority of calculations involving MPG require you to take the reciprocal first. Like, the "average" for average fuel economy of different cars is the harmonic mean. It's silly. Gah.
Imagine that you have an HIV test that is 95% accurate (false positive rate of 5%). and around 2% of tested population is infected with HIV. What is the probability that you actually have HIV when your test comes back positive? Is it 95%? Nope, actually just 29%. You do the test again.
Numbers from wiki about the paradox: https://en.wikipedia.org/wiki/False_positive_paradox
Strong profiling is not mathematically optimal for discovering rare malfeasors http://www.pnas.org/content/106/6/1716.full?sid=3bc684ec-b59...
https://arstechnica.com/tech-policy/2018/05/uk-police-say-92...
"A mathematically optimal strategy would be 'square-root biased sampling,' the geometric mean between strong profiling and uniform sampling, with secondary screenings distributed broadly, although not uniformly, over the population."
Those numbers are from 2007 - given that I learned this result in an entry-level undergraduate stats course, I'd like to think things are improving. But I'm not especially optimistic, and even if they are it still means generations of practicing doctors weren't taught this.
Would this be less of a paradox if we talked about water balloons? Potatoes, in real life, are 80% water [1].
Potatoes are a poor choice, most common salad vegetables are around 95%.
http://www.econtalk.org/odonohoe-on-potato-chips-and-salty-s...
Fascinating podcast about the economics of the (potato) chip industry.
Still an interesting question, but it feels a bit cheap to invoke a physical intuition that's completely irrelevant.
[1] https://en.wikipedia.org/wiki/File:Potato_paradox.svg
[2] https://bogleheads.org/w/images/thumb/8/84/Bond_-_Premium_-_...
If a bond has say a one-year term then it won’t lose 50% of its value if interest rates double from 1 to 2 percent because you still get your original full investment back after one year. It’s value instead drops by about 1%.
For a longer term example, a bond purchased for $1,000 and 1% interest rate with 30 years left is worth $776 if interest rates rise to 2%
Source: https://m.free-online-calculator-use.com/bond-value-calculat...
Let's say I have a bond that has a face value of $100, yields 1%, and matures in 1 year. It's value today is $99. Now interest rates go to 2%. That bond is now worth $98, because 1 year from now, at 2% interest, it will be worth $100.
But for longer term bonds (30 years, say), what you said can be true.
Duration: https://en.wikipedia.org/wiki/Bond_duration Convexity: https://en.wikipedia.org/wiki/Bond_convexity
It will come up anytime you have a secondary metric which is a percentage that you want to decrease but the total is the primary metric that people focus on.
One way to have 99% free users is to have 99 free users and one paid user.
Similarly, a way to have 98% free users is to have 98 free users and two paid users.
But GP didn't say "convert one free user to paid" or "acquire another paid user by other means." The number of paid users isn't changing: you will still only have one paid user.
So you have to cut both numbers in half to keep the same ratio. Instead of 98+2, you end up with 49+1 to have 98% free users.
1 / 100: 0.01
1 / 99: 0.01010101
That's just a bit more than a percent of that percent of change, and not a 100% change, as would be required for a 2:98 ratio.