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> 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox

I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another.

> 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood

This one, on the other hand, I don't like. Depending on a whole bunch of subjective definitions, a one-in-billion event can happen a million times a second or practically never or whatever else you choose.

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#16 is something video games taught me, particularly Path of Exile. In PoE resistance are a flat multiplier to incoming damage. Eg monster does 100 damage per attack and you have 60% resistance then you take 40 damage.

The interesting thing is that the higher your resistances the more effective each additional percentage point of resistance is.

Let's say a monster does 100 damage per attack.

If you have 0% resistance and increase it to 5%, then your incoming damage went from 100 to 95. You take 5% less damage than before.

If you have 75% resistance and increase it to 80%, then your incoming damage went from 25 to 20. You take 20% less damage than before.

It is pretty unintuitive until you realize that you need to focus on the remainder rather than the other part.

OK, let's assume you have 75% elemental resistance. You also take 15% reduced damage and 10% less damage. You have 5% chance to avoid elemental damage and 10% to dodge it and are under the effect of Elemental Equilibrium and Gluttony of Elements. How much better is it to just kill everything before it can kill you?

You joke, but PoE's use of stuff like 'increased/decreased' and 'more/less' to distinguish between additive and multiplicative calculations is one of the smartest game design decisions I've seen.

The game has a lot of seemingly arbitrary distinctions and concepts that you just have to learn over time, but once you actually learn them, the consistency of it all makes it very easy to handle the large amounts of complexity in the game.

It's completely unbearable going back to other games that just say crap like "+30% to x" without actually distinguishing between the different ways calculations can be done, forcing you to either experiment endlessly, look up every tiny thing on a community wiki, or just wing it.

It's a nice contrast to something like WoW where every time you get a new item you just chuck the item code into some ten million LOC simulator and fuck around with limiting permutations until it doesn't take 15 minutes to run, just to find out through some totally opaque process that you have 189 more dps. And then 2 months later you find out there was a bug in the simulation and the item you deleted 1.9 months ago was actually better.

A similar example with league of legends:

One point of resistance gives 1% effective extra health no matter how much you already have.

100 armor give 50% reduction (100/100+100) while 200 armor give 66% (100/100+200)

The percentage 50 -> 66% is shown ingame and players often think the value per point of armor drops.

What does actually happen is your effective bonus health changes from +100% to +200% and every additional point will be worth the same

Interestingly, the same logic applies to vaccination rates. Going from 0% to 5% vaccination has no impact on the course of the pandemic (except for those few vaccinated people, of course). Going from 75% to 80% has a much larger impact, and could stop the pandemic in its track (depending on R_0, and many other real-world complications of course).

(And the reason is just the same: what matters is the remainder.)

A similar but not quite the same mechanic is fuel economy.

Let's say you have two vehicles, both doing 10,000 miles per year. One gets 10 mpg and the other 50.

Would you rather upgrade the 10 mpg vehicle to 13 mpg, or the 50 mpg to 100 mpg?

Not only should you pick the former - you should pick the former even if you could upgrade the 50 mpg vehicle to run on nothing.

Yes, the fuel savings in the former case are larger than the initial fuel consumption in the latter case, but is it really unintuitive in practice? What I mean is that we generally pay for fuel per volume, not per mileage. Now, I am not sure about others, but I would always base my decision based on the money I'd save over a period of time, which in this case would require considering each vehicle's actual fuel consumption over that period of time.

It is the assumption that the two cars always make the same number od miles indepedently od the cost that is unusual and unintuitive.

If you think of it as a family that keeps obstinately driving both cars the same amount despite massive cost differences its weird but you could think of it as a mixed fleet of delivery or work trucks that are all needed regardless, the question is how you manage or prioritize upgrading them.

Yes, it doesn't explicitly state the rate or distribution of events. But it is a good reminder of what happens when your whatevers/second are pretty high - see the famous "One in a million is next Tuesday" [1]. "Rare" is soon if you roll the dice fast enough.

Any time your service has a high TPS, your API gets a lot of calls, a button in your app gets pressed a lot, ... this applies.

Critically, "a lot" is defined relative to your failure tolerance. It may actually be very fast or a lot, or not particularly fast but it really really needs to work.

It highlights the fallacy of equating "low probability" and "won't happen".

[1] https://docs.microsoft.com/en-us/archive/blogs/larryosterman...

Or everyone's newest favourite, the virus randomly mutating is "rare".

"Given the scale that Twitter is at, a one-in-a-million chance happens 500 times a day."

https://www.ted.com/talks/del_harvey_protecting_twitter_user...

> And I’ve seen some absolute doozies in my time – race conditions on MP machines where a non interlocked increment occurred (one variant of Michael Grier’s “i = i + 1” bug)

I could not find any info about that bug, anyone got a link or a source?

I assume that bug is referring to the fact that while i = i + 1 may look atomic to you as a human, in the computer it turns into

    Read i into register.
    Add one to that register.
    Write i back to the memory location.

And there's a window during that "add one to the register" where you can obviously have something jump in and write something else to that memory location.

What happens on your real processor is more complicated since this is going to relate to cache coherency between the processors, not directly writing RAM at that point, and that's a deep rabbit hole. I couldn't describe it all in detail anyhow. But I can observe it doesn't take much at all to turn that one cycle vulnerability into something with a larger target.

> "Rare" is soon if you roll the dice fast enough.

Indeed. One fun example is LHC[1], where the probability of a proton in a single bunch hitting a proton in the bunch going the opposite direction is on the order of 10^-21, yet due to huge number of protons per bunch and large number of bunches per second, it still results in ~10^9 collisions per second.

[1]: https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.lhc_...

16 is like the money hall problem. I understand the answer, the answer makes sense to me. And yet but when I think of it how I think of it initially ... it still makes no sense.

The way to make sense of this is not to think about the water weight but the solid matter. By changing the proportion of water from 99% to 98%, you're also doubling the proportion of solid mass from 1% to 2%.

> you're also doubling the proportion of solid mass from 1% to 2%.

And the final step is then, that the solid mass didn't change and therefore the liquid must be halved, instead of the solid doubled.

The limit case can be helpful here, a strawberry made of 100% water can be dehydrated to practically nothing and is still made of 100% water.

What you said makes no sense. If I'm only judged by how much money I have, then even if I have $0 I'm still a billionaire?

For me the mistake was 1:1 relation between percentage and weight which didn't remain true after weight loss but I thought it did

Easiest for me if thinking only in fractions and percentages, and realizing that the dry mass is a constant.

The obvious example is 99g water, 1g dry mass. Knowing that 1g cannot change, what do we need water to be to equal 98%? 49g.

*Monty Hall

The strawberries remind me of the pricing of long-term bonds.

Let’s say newly issued 100yr Treasuries pay a 1% coupon today, but tomorrow the coupon will be 2%, how much does the price of the older bond change?

The answer is a near 50% loss, simply because this is required to bump the yield of the older bond to 2%.

(If we include the discounted principal payment the exact answer becomes a 41.3% loss)

> > 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox

>I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another.

Comes up a lot lately because of vaccine effectiveness e.g. 95% is twice as effective as 90%.

and sunscreen (SPF) calculations...

Wait, how is 95% effective twice as effective as 90%?

It halves your risk.

90% is 1 in 10, 95% is 1 in 20.

just as going from 98% effectiveness to 99% effectiveness halves your risk (you go from 2% chance of falling ill to 1%)

its a common concept in games in which armor follows a linear formula (each point of armor is more effective than the last when calculating effective health)

Then there's the polynomial wizard.

It's probably more intuitive if you say that it's "half as ineffective"

I first saw this in a discussion of power supply efficiency. A 95% efficient supply generates ~half the heat that a 90% one does.

Here's my attempt to understand what's going on:

100g strawberry total weight where the 100% is made up of 99% water and 1% solid matter. 100g strawberry total weight where the 100g is made up of 99g water and 1g solid matter. 99/100 = 0.99

50g strawberry total weight where the 100% is made up of 98% water and 2% solid matter. 50g strawberry total weight where the 50g is made up of 49g water and 1g solid matter. 49/50 = 0.98

>> 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood

> This one, on the other hand, I don't like. Depending on a whole bunch of subjective definitions, a one-in-billion event can happen a million times a second or practically never or whatever else you choose.

I think this is about events happening to people, the number of people alive (and assuming they all communicate "miracle" occurrences"), and how many things they experience.

That is, if I understand if correctly it's not that you can choose a random number between one and a billion and run a CPU to randomly check numbers in that range as fast as possible and get lots of results in seconds, it's that based one how we have roughly 8 billion people all communicating events that things we consider "one in a billion" occurrences will be experienced about 8 times a month across the populate, and we'll all pretty much hear about it, which may not match with our expectations of how often we should see a "one in a billion" event reported.

Edit: Here's some relevant info from the paper "Methods for Studying Coincidences"[1]:

The Law of Truly Large Numbers. Succinctly put, the law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen. The point is that truly rare events, say events that occur only once in a million [as the mathematician Littlewood (1953) re- quired for an event to be surprising] are bound to be plentiful in a population of 250 million people. If a coin- cidence occurs to one person in a million each day, then we expect 250 occurrences a day and close to 100,000 such occurrences a year.

Going from a year to a lifetime and from the population of the United States to that of the world (5 billion at this writing), we can be absolutely sure that we will see incred- ibly remarkable events. When such events occur, they are often noted and recorded. If they happen to us or someone we know, it is hard to escape that spooky feeling.

A Double Lottery Winner. To illustrate the point, we review a front-page story in the New York Times on a "1 in 17 trillion" long shot, speaking of a woman who won the New Jersey lottery twice. The 1 in 17 trillion number is the correct answer to a not-very-relevant question. If you buy one ticket for exactly two New Jersey state lot- teries, this is the chance both would be winners. (The woman actually purchased multiple tickets repeatedly.)

We have already explored one facet of this problem in discussing the birthday problem. The important question is What is the chance that some person, out of all of the millions and millions of people who buy lottery tickets in the United States, hits a lottery twice in a lifetime? We must remember that many people buy multiple tickets on each of many lotteries.

Stephen Samuels and George McCabe of the Depart- ment of Statistics at Purdue University arrived at some relevant calculations. They called the event "practically a sure thing," calculating that it is better than even odds to have a double winner in seven years someplace in the United States. It is better than 1in 30 that there is a double winner in a four-month period-the time between win- nings of the New Jersey woman.

1: https://www.gwern.net/docs/statistics/bias/1989-diaconis.pdf

I agree with the parent that the cited "fact" is sort of questionable (along with some other things on the site even though I really enjoy it overall) because of ambiguity in definitions, assumptions, and so forth.

However, the law of truly large numbers, as you frame it, is something you experience firsthand working in high level severity hospital settings in large metro areas. There's a large enough hospital catchment area that you start to see, on a fairly regular basis, the medical outcomes of all the bizarre and unbelievable things that happen rarely to any given person. It gets to a point it's difficult to know how to explain because the details of each case would be potentially identifying given how strange they are. And yet something happens all the time. Maybe not that one thing, but something of similar impact. It gives you a distorted sense of risk.

>https://en.wikipedia.org/wiki/Potato_paradox

The Wikipedia link above says:

> Fred brings home 100 kg of potatoes, which (being purely mathematical potatoes) consist of 99% water. He then leaves them outside overnight so that they consist of 98% water. What is their new weight? The surprising answer is 50 kg.

It annoys me when mass is used interchangeably with force (weight), so I went to the Wikipedia source, and the source is accurate in using units of force throughout.

https://web.archive.org/web/20140202214723/http://www.davidd...

Wonder why the person that wrote the Wikipedia article changed it up when it is supposed to be a direct quote.

In this situation, the mass and weight are proportional and irrelevant to the problem. Other than proper respect of units, why would it really matter? I would agree that using mass+kg would remain correct and be less unusual, but it doesn't matter a lot.

It does not matter, it is just a pet peeve of mine. Might be due psychological trauma from when I was a kid and arguing with an older cousin about how pounds and kilograms are not units of the same thing, and the older cousin "winning" the argument in the eyes of the elders because the cousin was quite a few years older than me and considered to be smart in school.

From what I remember from intro physics, we distinguished between pounds and pounds force, the latter having the 32ft/s^2 multiplied in.

And wikipedia seems to agree with me, see pound (mass) vs pound (force).

Oh wow, learning a lot today. I was taught in the US that pounds are a measure of force, and that is how it was always used in physics problems.

As a European I learned in metric. When I first learned pounds, it was as the imperial system's equivalent of grams and a conversion factor was given. Force in physics class was taught in Newtons (kg*m/s^2).

Pounds as a mass unit are perverse enough. Things like pound force and psi (pounds per square inch) were used only to make fun of old mechanics papers and textbooks. Also, btu. It is quite amazing actually that someone would see the SI and think “no, too simple; I’ll keep my pounds, ounces, inches, and feet”.

Anyway, yes, the proper unit of force is the Newton.

I think there is some weird dual usage that makes them either mass or weight depending on the context. For example, torque is in ft•lbs or N•m so the pounds there are lbs force.

Though checking wikipedia again, it actually specifies that torque is measured in as lbf•ft. I take that to mean that 1 ft•lb is the torque of 1/2 oz (1/32 of an lb) at 1 foot. I expect that's a test question almost everyone would get wrong, myself included.

https://en.m.wikipedia.org/wiki/Pound-foot_(torque)

Think like an ordinary person. You know, an ordinary person who would say that they weigh about 80 kilograms. Only science nerds would say that they mass about 80 kilograms, or that they weigh about 785 Newtons. Similarly, anyone who's used to living with US customary (or Imperial) units understands that a pound of force is what a pound of mass weighs and would see no reason, under any circumstances, why anyone would want to divide the gravitational acceleration out of a pound-foot to arrive at a "real" torque value. When the pound value is expressing a weight-equivalent force, that force is the force of a pound under normal gravitational acceleration at or near the surface of the Earth.

In engineering school (in the US), we used pounds mass (lbm) as the unit of mass, and pounds force (lbf) as the unit of force.

To me (Netherlands) a pound is simply 0.5kg. Force is expressed in Newton.

I really don’t like that example because it makes no sense. In no logical circumstance could the potatoes dehydrate so quickly when left out over a single night.

And they also don't consist of 99% water. That is why they called them "purely mathematical potatoes" and they could've chosen any type of fruit or vegetable. Heck, I'm just waiting for a car analogy now!

Brake fluid anyone?

But that 99% water simplification is needed for the purpose of the exercise.

The left out overnight is basically an absurd statement that is intended to confuse and not really related to the actual question.

The statement could be “left in a dry environment until” or simply “left to dry a few weeks”

You definitely could write that but it wouldn't change anything and you could make the same "argument" you are making now. "The left to dry a few weeks is basically an absurd statement that is intended to confuse" and it still wouldn't be true. It's not intended to confuse at all. It's intended to get the point across that you let this imaginary thing dry from 99% to 98%. They could've said "sponge" and let it out to dry any number of minutes. The point isn't to make a 100% accurate example of the drying properties of any actual 'thing'. They just needed something that people intuitively know "has water content" and that "can dry".

Wow by the downvotes I learned people react extremely negatively to any critique of math word problems. Spherical cows and 99% water potatoes are fine, those over simplifications are required for the analysis.

Saying Leaving the potatoes out overnight to imply they halved in mass, sounds as reasonable as “the potatoes were in the ground for 12 hours and then doubled in amount, what percent of water are they now?” It’s so gratuitous and requires ignoring all other laws of physics, while the goal of the spherical cow type simplification is to only ignore a few key challenging ones.

You probably won’t be thrilled by the spherical cows in the nearby pasture then.

Do spherical cows dream of mathematical potatoes?

Spherical cows, naturally, dream of spherical potatoes in a vacuum whilst grazing next to spherical chickens.

Honest question to know how others think.

It doesn't really matter, does it? The rate of evaporation is irrelevant to the problem. Mr. Potato could have waited a year, or dried them on the Uyuni salt plains.

Why do you care? Would this distraction affect your ability to solve the problem?

> Wonder why the person that wrote the Wikipedia article changed it up when it is supposed to be a direct quote.

They likely changed it from lb to kg because that would be more friendly to an international audience, without realizing that lb is a measure of weight and kg is a measure of mass. Therefore they didn't know to change "weight" to "mass".

Oh yes, that would make sense.

I have been to the US quite regularly, and been living in the UK for a number of years, and I have never seen someone using pounds as a unit of force instead of a unit of mass meaning roughly 500g, give or take. Second meaning was about 1.20€.

FWIW, the pound is a proper unit of mass: https://en.m.wikipedia.org/wiki/Pound_(mass) .

I'm a civil engineer. In statute terms pound is the unit of force and slug is the unit of mass. That might have colored my thinking.

The difference between weight and mass is domain specific to physics.

I actually get annoyed at people who are pedantic about these things. Precision is important in some conversations, but just elitist in other.

Anyway, the term “weight” to refer to mass outdates its use as a force - its only since Newton that we distinguish the two, after all.

You're being overly pedantic (and I would argue actually incorrect). Kilograms and pounds are both referred to as "weight" in general conversation and nobody is going to be confused by this.

Go to any supermarket in a country that uses the metric system, potatoes will be sold by the kilogram - it's the natural way to phrase this outside of America.

In a physics context the definition of kilogram might be specifically mass, with newtons referring to weight/force. But words can have different meanings outside of technical contexts.

If you go to a metric country, and ask someone how much they "weigh", approximately zero people will say "x newtons", they will say "x kilograms" (or "x pounds" still in a lot of commonwealth countries if we're being pedantic).

Although the more I think about this the more I think the difference between technical and colloquial is actually that "weight" in colloquial use refers to mass, because force is not commonly relevant.

Yes, in colloquial use weight refers to mass _most_ of the time. But can also refer to inertia or mass. Or be used metaphorically.

"Weight" historically referred to mass, in common speech dating back forever. It’s the Germanic word which has been used throughout the history of English, whereas "mass" comes from Latin via French, like 5–6 centuries ago. The two words are almost exact synonyms, in historical/colloquial use.

Both historically and today, a "pound" (Roman libra) is a unit of mass. People use a pound-force as a unit of force only in somewhat specialized contexts.

At some point in the relatively recent past, someone (not sure who) decided that we needed to have 2 separate words for mass vs. force, and we should keep the Latin word for mass and use the Germanic word to mean force.

Now pedantic people are constantly insisting that using the standard English word weight to mean mass is "wrong".

Actually in the past "weight" or the Latin "pondus" (=> pound) always referred correctly to what is now named "mass".

When someone mentioned "weight" just in a qualitative way, as a burden, they might have thought at the force that presses someone down, but whenever they referred to weight in a quantitative way, they referred to the weight as measured with a weighing scale, which gives the ratio between the mass of the weighed object and the mass of a standard weight, independently of the local acceleration of gravity.

Methods that measure the force of gravity and then the mass is computed from the measured force, i.e. with the force measured either mechanically with springs or electrically, have appeared only very recently.

The distinction between force of weight and mass became important only since Newton, who used "quantity of matter" for what was renamed later to the more convenient shorter word "mass".

Perhaps it would have been better to retain the traditional words like weight and its correspondents in all other languages with the meaning of "mass", because this meaning has been used during more than 5 millennia and use a new word, e.g. gravitational force, for the force of weight, because we need to speak about this force much more seldom than about the mass of something.

> Actually in the past "weight" [...] always referred correctly to what is now named "mass".

That’s the same thing I just said. Why add “actually” in front? Yes, weight was historically measured with balance scales.

I guess I should have been clearer that the term “mass” as used in physics only dates from 3 centuries ago (from Newton), and did not historically mean weight in Latin. (Mass comes from Latin via French for lump of dough.)

You are right, I have misunderstood what you have said, because it indeed looked like if "mass" would have been some traditional word having anything to do in any language with what are now called "weight" and "mass" instead of a recent post-Newton word choice for naming one of the 2 quantities, while keeping the old names for the other.

I still think that the choice of which of the 2 should get a new name was bad, because the traditional quantitative meaning almost always referred to what is now called "mass"(with extremely few exceptions such when somebody would be described as so strong as to be able to lift a certain weight).

When I buy potatoes I am interested to buy 10 kg mass of potatoes, not the amount or potatoes of which has a gravity force of 98,1N. On mars you need to eat 10 kg of potatoes a week, not the amount of which has 98,1N gravity force.

The scales in the supermarket on earth automatically convert the weight into mass for my convenience by applying a constant factor of 1/9,81.

I am hardly far away enough from earth that the constant changed, so I did not need to distinguish between the two measures so far. When carrying the potatoes home I just use the mass of 10 kg as a proxy for the force I need.

And to determine the increased breaking distance of my car, I need to know the mass again.

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