Not exactly the same, but I stumbled on another seemingly confusing math reality a little while back. It's probably obvious for places that measure their mileage in gallons/100 miles or kms or whatnot, but for MPG:
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.
A similar one I enjoy: you're driving two miles. After one mile, you've averaged 30MPH. How fast do you need to drive the second mile to have a trip average of 60MPH?
Answer: impossible. You'd have to travel the second mile at infinite speed to achieve this.
If you drive the first mile at 30 MPH and the second at 90 MPH you averaged 60 MPH ((30+90)/2). What is the math that gets "infinite speed" as the answer?
You're averaging over distance, but to calculate the average speed you need to average over time. If you drove the first mile at 30 MPH and the second at 90 MPH, you'd spend 120 seconds on the first mile and 40 seconds on the second mile, thus giving an average speed of
Helps to think about it like this: if you averaged 60 MPH on a two mile drive, your trip would last two minutes. Since your first mile was driven at 30 MPH you've already been driving for exactly two minutes. The only way to hit an average of 60 MPH for the whole trip is if the second mile is instantaneous.
Put it this way: If you were to average 60 mph over 2 miles, you would travel those two miles in 2 minutes.
However, 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.
For your method to work you need to add the time spent at speed, not the miles covered.
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.
This relates to another mathematical "oddity". Some regions in the EU measure fuel economy with km/l, while others measure in l/100km. Years ago, when the EU first released fleet fuel economy standards, there was a bit of confusion as some manufacturers averaged the fuel economy of their fleet and then converted to the appropriate units. Which gives a different result than converting to the appropriate units and then averaging.
As much as I argue that fuel/distance is the correct way to measure things, given that the vast majority of people keep the distance they travel fixed as the amount of fuel and its cost fluctuates, it is a bit easier to do certain math with the distance/fuel metric.
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.
Most cars don't tell you how many gallons are left in the car, just approx how much of a full tank. So mpg doesn't help much in your example, miles per full tank is a useful number though and from there you can calculate that if you have half tank you only have 300/2=150 miles left. Though most cars will tell you the exact remaining distance to empty in the instrument cluster already so there is no need to do this mental arithmetic anyway.
As the owner of a very basic 2003 car, I can assure you that the mental arithmetic remains. I know that each 8th of my tank on the gauge is worth ~50 miles, entirely dependent on how efficiently I'm driving.
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.
It's only the same amount of fuel savings if you burn the same amount of fuel, right? The amount of fuel saved between 10mpg -> 12mpg may be roughly equal to an improvement of 30mpg -> 60mpg, but being able to go e.g. 600 miles (one 10-gal tank) vs 300 miles could be the difference between buying one tank or two (or: 1000 or 2000 over your car's lifetime).
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?
It's a simple illustrative paradox, but the world is complicated.
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.
This is indeed surprising! For those like me who need a bit more detail before this clicks:
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.
One faces a similar problem with thinking about how much time is saved to go a certain distance: velocity is measured in meters per second, not "slowness" in seconds per meter.
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.
Fuel economy should be listed in fuel volume per distance. (I assume it is not partially due to automobile/petroleum industry pressure.)
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).
Miles per gallon is a horrendous unit of measurement for this exact reason. Trips are of fixed length, and you generally want to calculate fuel usage for those trips. MPG is useful for comparing what sort of trips you can take with a fixed amount of fuel, which describes approximately zero real-world situations.
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.
That assumes you do the same millage but the people might drive further if they get 30mpg rather than 10mpg making the 30-60 saving greater than the 10-12. Indeed if you assume economics 101 that people consume more if the good is cheaper, that would be the case.
I believe this is the implicit main argument behind the "rational aid" movement: it is very hard to save and gather resources, but very easy to squander them. Similarily it's very hard use rational thought to marginally improve the human condition, but its very easy to worsen the human condition by not using rational thought for just a split second...
The UK isn’t exactly metric. Perhaps imperial measurements like Brexit will die out.
So many measurements are daft units: road distances, trousers, the nutty papers still use Fahrenheit.
Dry goods are in 450g units, and milk is sold by pint in ml.
No one is truly metric. Even australia, the only country I know that does cm for screen sizes, still uses inches for car wheels just like everyone else.
Another somewhat similar (in that the result is unintuitive) problem about false positives:
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.
Same issue comes up when you discuss effectiveness of ethnic profiling of Muslims/brown skinned people in airports. Sam Harris was unable to wrap his head around this and was sticking to his common sense intuition.
And more recently, UK police have discovered that surveilling crowds with facial recognition software produces almost entirely false positives, even if the software is supposedly quite accurate at picking out criminals. Not that they care or plan to stop using it, of course.
All you're saying is that any Muslim picked up is more likely to not be a criminal, which seems expected. The problem with "strong sampling", is, perhaps obviously, that they focus too strongly on the 'brown skinned people' and therefore might miss other criminals. The article you link seems to suggest some profiling is good, but that strong profiling AND random sampling are equally inefficient.
"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."
Depressingly, this is apparently unfamiliar even to doctors who actually face this exact situation on a regular basis. Given multiple choice tests, they get the right rate at (or a bit below) random chance, and a large plurality think that the test accuracy equals the chance that a positive result is correct.
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.
Gerd Gigerenzer has a good book called Reckoning With Risk that talks about this, and gives several real examples of people taking action after being given information by medical staff.
I'll bet using potatoes does makes the the question harder for people. However simple the math, it runs up against the intuition "potatoes are full of starchy solids, I'll bet lowering their water weight one point wouldn't be that noticeable a change". Heck, I'll bet this is made harder by exposure to baked potatoes. The actual study on water loss via baking potatoes is paywalled, sadly, but discussion on some fitness forums says that there's a >5% weight loss from baking - which would constitute a 1-point change in water weight.
Still an interesting question, but it feels a bit cheap to invoke a physical intuition that's completely irrelevant.
Reminds me of how bond yields work. If the interest rate guess from 1% to 2% on an existing bond, the bond itself has to lose 50% of its value. I think the potato story is somehow less intuitive.
That’s true only if the bond has an “infinite” term. Meaning it pays interest forever but you never get your initial principal back
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%
This is true in limited cases. Changes in bond values in relation to interest rate changes is much more involved than this. The first derivative is known as "duration" while the second derivative is known as "convexity". The size and timing of payments will vary from bond to bond, which heavily affects interest rate risk:
The potato paradox disappears entirely with the right illustration [1]. Something similar (though not quite so clear as paradox's grid) for bonds: [2].
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.
Profiling code. If you have a hotspot that takes 99% of the time and optimize it until your whole program is twice as fast, now that hotspot takes 98% of the time.
Possibly. If you have 99% free users and you want to get to 98% free users by convincing free users to stop using your platform, then you need to lose half of your users.
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.
What helped me understand this is to look at a specific example.
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.
The key phrase was by convincing free users to quit, i.e. the number of paying users remains unchanged but you want to improve the ratio of paying to free. To double the ratio, you have to halve the total (being sure that no paying customers quit).
This isn't the intuitive result? If one part in 100 of the potatoes is solid, and you want to increase that to 2 parts in 100 that's equivalent to 1 part in 50...
Almost everyone I know could not solve it correctly without a lot of hints. Even when you say: „It is not the answer you first think it is“. I like to use it when someone feels too smart and won’t acknowledge that he might be wrong about something. For example, a friend was arguing that the moon landing didn’t happen because the capsule was too small to hold the needed fuel. I asked him the paradox, he got it wrong. I followed with: „how can you believe to be right about rocket science, when you weren’t able to get this simple question right?“
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.