Simple, yet counterintuitive mathematics | Why numbers don't always mean what you think

[@factsopinionsandinterestin6832]

Ironically, we intuitively guessed that the guy on the right saved more money on gas, because you wouldn't have asked otherwise...

[@howard5992]

It was also a case of the question being presented in a way that is somewhat misleading. The guy on the right is spending more (because his car burns more fuel). A 20% reduction in a large number can be much more than a 40% reduction in a small number.

[@AdmiralJota]

For me, the fact that he started out telling us fuel efficiencies but then switched and asked us a question about costs was a flag for my brain to invert how I was thinking about it. In contrast, the strawberry question didn't have that switch (both parts were about the percentage of the water), so the answer was less intuitive to me than the first one.

[@ViratKohli-jj3wj]

Haters are gonna hate

[@NMPshadow]

I just realized, why I fell for the trap in the first place. Where is the efficiency counted in km/l instead of l/km. Even the numbers (range from 3 to 14) look more realistic in a l/km environment. (At least in Germany, if someone asks how much your car needs, you tell the l/km, not the other way round)

[@57thorns]

@@NMPshadow Yellow car is pretty bad, green car is bad, blue and grey cars are probably driven at high speed on Autobahn.

[@TheOnlyJura]

Damn, I can't believe you actually wrapped a cable around the globe just to make this video!

[@DirtyPhlegm]

lmao

[@georhodiumgeo9827]

Worse than that he plugged it back into itself with a USB cable! LOL.

[@TheGoukaruma]

The lifting part is the tricky one.

[@FastEddy1959]

I was wondering why there was a cord across my lawn!

[@piesho]

Some people go the extra mile

[@freakydeaky1435]

Oh man, the extension cable example made going up a belt size seem so much worse :(

[@Brosylen]

On the contrary, I find it makes it less worse : On my belt, the holes are 3 cm apart (≈ 1.2 in). If I gain one size on it, meaning my circonference increased by 3 cm, it means my radius only increased by 3/2π ≈ 0,48 cm (≈ 0.19 in). I feel better if I tell myself that this gained sized just means I got less than a half cm increase on my radius. I actually feel worse when I remind myself that it means that I (having a circonference of about 100 cm) have gained a strip of about 100×0,48 = 48 cm² on any slice of my belly...

[@kkounal974]

@@Brosylen really goes to show you how counter intuitive volume can be huh?

[@jebeda]

The metric measurement for gas consumption is L/100km, so the green car (lower left) gets 9.44 L/100km, and the yellow (top left) gets 6.71 L/100km, thus saving 2.72 L/100km. The grey car (bottom right) gets 29.4 L/100km and the blue car (top right) gets 23.5 L/100km, thus saving 5.8 L/100km if you switch. Measuring gas consumption in a unit that actually measures CONSUMPTION of gas per amount driven is much more meaningful for these types of comparisons. MPG (or km/L) is not actually a "consumption" but more like a "range" measurement.

[@chrissim4386]

@M Detlef Actually you´re wrong. Consumption means how much does it take to achieve something. That means liters per distance, e.g. 5l/100km (equal with gallons and miles). But there might be other terms than consumption that turn it around. I think efficiency could be seen as one. Meaning fuel efficiency should be displayed as 15km/l, because it tells how much you gain out of certain amout of ressources. You gain 15km per liter.

[@ludiProjects]

@M Detlef By definition, consumption is the amount of fuel required to perform a unit task (drive a kilometer). So you're wrong

[@harutodoroki9918]

but still the left guy uses less money for gas right? the right guy just saved more with the purchase

[@jensraab2902]

Ha, I was confused by Chris Sim and Ludicrous Fun attacking the OP to be wrong when he clearly wasn't - until I realized that they were responding to a comment that has been deleted. Should go off the internet and get some fresh air...

[@simanto86]

@@jensraab2902 I was wandering what was wrong with Chirs Shim and Ludicrous Fun until I read your comment...

[@jujeremy]

The first problem shows why measuring fuel efficiency in L/100km is much more useful. Since the units are volume per distance rather than the reciprocal as with mpg, you MULTIPLY your distance by fuel efficiency to compute the volume of fuel consumed. So, if you rephrased this question for someone not using wonky hillbilly units, it would read "One person trades in their 9.4 L/100km car for a 6.7, while the other trades in their 29.4 for a 23.5. Who buys less fuel compared to before?" The answer is then intuitive and obvious.

[@alexcondos9897]

RIP hillbillys

[@PeterAuto1]

I was looking for this comment

[@selenasilverstep7981]

insert America measurement units meme here

[@robinw77]

Yeah "km/L" looked weird too. And since moving to mainland Europe, L/100km is more intuitive now.

[@johngaltline9933]

And yet, anyone that routinely uses many different units has little trouble swapping between them. More problems are cause by people thinking the world is sunshine, rainbows, and same arbitrary units than by there being any fault in those units. It doesn't matter if you use donkey tails per bathtub or fractions of bath tub per donkey tail, as long as you can do the math. In the above question, sure, it takes one less step if you measure fuel per distance instead of distance per fuel, but it would likewise add a step if the question asked, say, which driver can travel further on vacation on the fuel saved. People need to get over thinking any unit that is not metric is some how inferior... especially in a question that provided the metric equivalents in the first place. Should also probably stop building their buildings in weird fractions of meters that just happen to work out to even numbers of feet.

[@timchallenge]

Just discovered this channel and I'm hooked!

[@zachstar]

glad you found it!

[@thosekids321]

Same here

[@chriswebster24]

I found this channel so early on that I’ve been able to watch every video, thousands of times each, and I’ve even memorized most of them word for word. Fortunately, however, unlike you, I’m not even the slightest bit hooked, thank God. That would totally suck.

[@duartecastro623]

Same here...

[@westinbye]

Haha me too! Hour 4 and still going

[@HelplessGazellle]

Hey, you should watch your youtube statistics to see if your views go up over the next few weeks. The main thing stopping me from watching your videos is other time commitments. But now that schools/colleges are being canceled, I think a lot of people will have enough free time to watch more of your videos. I know I sure will. Thanks for making great content. Your job is my dream job.

[@zachstar]

Yeah definitely will keep an eye on the views! You're right and considering most of the audience is in school then wouldn't surprise me to see a bit of a spike over these next few weeks.

[@spaceyonyoutube]

@@zachstar Ive been watching your videos recently and im defo gonna watch more. ima nerd

[@zachstar]

@@spaceyonyoutube haha we all are, glad you've enjoyed!

[@explosionmental2781]

SO TRUE keep up the good and old work c:

[@arguel53]

Det

[@ssifr3331]

It's amazing how in the third example the original radius doesn't matter at all.

[@ca-ke9493]

Having it as a sphere/earth obscutates the fact that the whole situation linear in nature. If we were to ask the same question by drawing a circle on a piece of paper, the intuition might be different

[@irrelevant_noob]

@@ca-ke9493 *obfuscates ;-)

[@unclecreepy4185]

There was a sports show called “Numbers Never Lie” where they used a lot of stats to talk about games and make predictions, etc. But they were wrong so many times because of the simple fact that while numbers never lie, people do. They would constantly misinterpreted or misunderstand the stats and numbers. This is especially true when dealing with a small sample size where adding 1 or 2 can double the percentage.

[@NibbleMeTwice]

Clearly, they didn't watch Moneyball.

[@cerwe8861]

A mother is 21 years older than her Child. In 6 years, the Child will be 5 times younger than the mother. Where is the father?

[@elfro1237]

Well I know it was t the question but negative 2 and 19

[@cerwe8861]

@@elfro1237 the mother is 20+¼ years old, the child -¾ which in month is -9 month... If you know a bit of biology you know where the father is

[@elfro1237]

Dbzfan _21 wait yeah I’m stupid I forgot that the mother would by 25 not 20 sorry

[@mattc3581]

@@elfro1237 Though 19 and -1 would end up as 25 and 5 which also works, in which scenario anyone could be the father, they might not even have met yet.

[@LucyTheBox]

He's at the grocery store, getting milk

[@vidhound]

I’ve told people the circumference puzzle before, and once you do the math, you figure out that of course it is the non-intuitive answer. But I also am looking for ways to visualize it that make it more intuitive. One way that I think makes it easier to picture is if you imagine the earth being a cube rather than a sphere. It is then much easier to picture what adding 6 feet does to the height of the cable, since it gets 1.5 feet on each of its four sides.

[@descuddlebat]

Simple intuition for the last ace thing: The probability for a position to contain the last ace is the probability that it contains an ace, times the probability there is no ace after it. The former a constant 1/13 (4 aces / 52 cards), the latter is 1 only if there are no cards after this one.

[@KarlFranzOtto]

I was thinking the same, but he explained it so complex XD.

[@takyc7883]

You should do a "How easy it is to lie with statistics Part 2" I woiuld love that, and im sure many others would! You could use real world political examples too.

[@kingofgrim4761]

Well if you use real world political examples, then there’s gonna be a LOT of dislikes by the right and left parties 💀

[@takyc7883]

@@kingofgrim4761 Yes, with very petty arguments in the comments section with no evidence/explanation!

[@stephenkamenar]

the first example is so confusing because the guy on the left IS saving more on gas in terms of percent. He's saving 40% instead of the other guy saving only 25%. but ofc the guy on the left's gas bill was already so low saving 40% on it doesn't matter that much. very interesting though, fooled me! o:

[@onebronx]

Yep, this is basically what a "law of a diminishing marginal return" and a "low base effect" are all about.

[@jp4431]

Technically it's 29% (1-(1/35)/(1/25)) and 20% (1-(1/10)/(1/8)) but I get your point, though the question is who will save more gas in absolute terms, not relative terms.

[@dascandy]

That example is great for realizing why insulating your house makes for much more difference on completely uninsulated houses, versus partially insulated ones. Yes, you still leak 70% of the total energy, but that saves you 30% of the energy cost. Going from 80% to 99% only saves 19% of the energy cost - which saves you 95% of your bill, but that bill was peanuts to start with.

[@kcbsuiejd]

yeah, that threw me off too. thats why its important to talk about weather you mean total or relative difference

[@howard5992]

The driver with the poorer mileage is using much more gas to start with. So we can simply say that a small percent of a very large number can be much greater than a large percent of a very small number. That is very easy to visualize as a general concept.

[@stephenkamenar]

the strawberry example is the same as stock prices. if a stock drops 99% vs 98% the difference in price between the two is 100%. you'd have twice as much $ if it only dropped 98% instead of 99%

[@michelestraface98]

i wonder in what kind of markets do you invest....

[@jwil4905]

@@michelestraface98 How does that matter to his (correct) point?

[@bguy510]

@@jwil4905 le joke

[@juzoli]

Better way to thing about the “cable around” the world question INTUITIVELY, is the following: Imagine Earth is a square (or cube). Shift the cable by 1 feet on 1 side of the square by adding 1-1 feet to the 2 sides. So 1/4 of the cable will be 1ft above Earth. Repeat it on all 4 sides. So we added 8 feet, and lifted the cable by 1 feet all around. This is a good approximation for the case of circle/sphere too.

[@PapaFlammy69]

mah boi sneakin'

[@ultragamer4465]

@Bob Trenwith no

[@arnavrawat9864]

@Bob Trenwith no this is patrick

[@frogstereighteeng5499]

Father Flammy, we love you.

[@trebmaster]

Pulled a sneaky on ya

[@mrsatire9475]

The neighbor on the left sees a cost savings of 29%, while the neighbor on the right only sees a cost savings of 20%, however, the later is wasting so much money that his reduction is about as equal as the former's entire bill.

[@digitig]

The way we solved the cable around the Earth one when I was at school was "We're not told the radius of the Earth, so presumably it doesn't matter. That means we can set it to zero, and the answer is simply the radius of the circle whose circumference is the length of the cable."

[@ryanburkett949]

Gotta be careful with that line of reasoning. If you were ever dividing by a measurement that didn't matter you cannot set it to 0. Just any positive number. For example, if you had y/x = 20/x. You could say, well we don't need to know the value of x to solve for y and since it doesn't matter lets just set it to 0. But then the equation becomes undefined. I always replace things like that with "1" instead.

[@digitig]

Ryan Burkett If the measurement didn't matter I wouldn't be dividing by it. Of course, the line of reasoning only works for puzzles, not for the real world, because in puzzles you can assume you have the information you need.

[@ryanburkett949]

@@digitig True.

[@ryanbell3704]

@@ryanburkett949 or you could just work with x=(something positive but really really small that’s close to 0) like 1/abazillion or .000000000000000001

[@moonman239]

The reasoning I went with was: 2 * pi * r (new radius) = 2 * pi * re (earth radius) + 6 r = (2 * pi * re + 6) / 2pi = re + 6/(2pi) = re + 3/pi r = re + 3/pi In other words, (r - re) = 3/pi = 0.95. Being pedantic, we should probably round that to 1 to account for significant figures, though.

[@AccAZ09]

The card game actually illustrates really well the Monty Hall riddle, especially when downed to 3 cards only

[@markschultz2897]

The reason why it doesn't work is because the cost depends on the amount of gasoline burned, which is part of the denominator, so you first have to switch from kilometers per liter to liters per kilometer so that the amount of gasoline burned is in the numerator, then you can do a simple calculation to get the right answer.

[@Reydriel]

It's unintuitive because it involves reciprocals, basically.

[@toyfabrik2993]

Here's some even more counterintuitive maths: Since he apparently got some 40,000 km of cable lying around (and he wouldn't have lied to us about the actual lenght of the cable shown in the video??) i did a rough calculation: Here in Europe, for 100 m of double stranded copper cable, the cheapest variety, the cost would be about 25 € (or 30 Dollars). So it's about 300 Dollars for 1 km of cable. Multiplied by 40,000 km, this is about 12 million Dollars worth of cable (!) All spent for a single KZbin video. This man is a true hero of the internet! ^^

[@DominicLeDuc]

I never never comment, but Zach your videos are so crisp and clear, with beautiful animations and intuitive explanations, I had to let you know. I'm about to finish undergrad and am so glad I found these, they'll keep me learning beyond school. Keep it up!

[@zachstar]

Thanks Dominic!

[@immabhonest1]

Zach I'm going to be honest with you: this is way more interesting than most math in school. You're an excellent explainer and I love ur vids

[@YumekuiNeru]

this is math you learn in school

[@obibellowme]

YumekuiNeru well most schools don’t present it in such a clear and interesting way

[@zachstar]

haha glad you enjoyed!

[@Reydriel]

@@zachstar The card problem you presented (guessing the last ace in a deck) is actually related to some of the permutations/combinations problems I learnt back in highschool haha. Basically, you can count the total number of permutations (n!) by multiplying the individual permutations of each type of object in the set together with a combination (choose function) number

[@xj-vn4eo]

here's something I want to add: They are not the same math you learn indeed. For the video has the following two properties: 1) it is good for giving you stimulation, to get you excited. 2) it is not so good for in-depth study, and for giving you underlying principles behind those counter-intuitive examples. When you, or let's say I learn math in school, I get to see many underlying principles. For example the linearity in the cable example; combinatorics in the card one. Say the card trick. What if the problem changes. If you recognize the underlying principles of combinatorics, you will be able to solve it. But simply watching this video won't get you that far. I'm just trying to recognize the video as it is. Not trying to overstate or understate it. (I did like this video, for its being interesting and relatively concise.) And since we are on the topics, I do think the key is to present sufficient real world examples and in a fun way, which is what some of the programs do, at least what I have experienced.

[@N4w4k]

The metric way of expressing fuel economy is in liter/km not km/liter. Converting into those units makes the first exercice not counter-intuitive at all! :)

[@mcdemoura]

It depends on the country. In Brazil, for instance, we only use km/liter, never the other way around. There's no right or wrong way of expressing fuel consumption/economy/efficiency.

[@ssifr3331]

@@mcdemoura In Indonesia it's also km/l

[@WaffleAbuser]

Those are both valid units, none is more correct than the other.

[@DanteKG.]

In my country fuel efficiency is stated as liters per 100km. I must admit that km/l makes much more sense

[@zachstar]

I very recently found out that some countries use L/km (or really L/100km), I had never known that before but yes in terms of the question it absolutely changes the intuition.

[@circledline3880]

Thank god he said he was kidding at 7:47 or else I would've believed him.

[@randomness9960]

I’d be hitting whichever casino has this game and winning me some money 💵 💵💵

[@irrelevant_noob]

*7:55

[@circledline3880]

To clarify, I picked 7:47 because 7:55 doesn't give much context to the clip. Also yes, when it comes to the probability of cards, I am not smart, but I was also playing up the stupidity for the sake of the joke.

[@irrelevant_noob]

@@circledline3880 understandable... i guess we just have different opinions on what is enough (and too much) context. :-)

[@circledline3880]

@@irrelevant_noob B^)

[@ir2001]

I appreciate the growth of your channel along with the content over time. Keep it up 👍🏻

[@davidjames1684]

The gas question is quite easy since a gas guzzler is already consuming lots of gas, so ANY improvement in that is likely to save more money than an already fairly fuel efficient car. An easy way to see this is to compare a 10 MPG car to a 100 MPG scooter. Drive 100 miles so the scooter uses 1 gallon and the car uses 10 gallons. Let's say each gets improved by 10% so the scooter can go 110 miles on 1 gallon and the car can go 11 miles on a gallon. Any identical improvement to both would have the gas guzzler saving 10 times as much on gasoline total price. The improved car will use about 9 gallons of gas and the improved scooter will use about 0.9 gallons. A saving of about 1 gallon vs. about 0.1 gallons which is a big difference.

[@Ulkomaalainen]

The main thing about the "cable around the earth" part is indeed psychology - or at least our frame of reference. We are used to driving 100 miles (or kilometers) to our hometown, and it really isn't that far away if compared to, say, international travel. Unless you live in Vatican City, but then you'd probably not know how to use the internet and wouldn't be here. However in height, we probably don't go above 10km for the level at which a plane flies (or, talking orders of magnitude, the Mount Everest). We just don't have anything to do with the 6000+ kilometers of Earth before it reaches what we call the floor. If the question was "we have a circle with a radius of 6000km and make it longer by 1m, it will raise a couple of cm" ("longer by a yard it will raise a couple of inches") nobody would be surprised at all. The cable is longer by a negligible amount in context, so the radius also increases by a negligible amount.

[@kateba]

I thought about the ace question like this: For every position, there is a 1/13 chance that it is an ace, so this is the unimportant part. So the important question is, if it is an ace, will it be the last one. This chance increases the further you go in the deck, so 52 (where the second chance is 1) has the highest odds of winning

[@RC32Smiths01]

Hey man, it always great to get an enlightening on what it means to have numbers in interpretation! I was wondering if you could do a video on Number theory actually

[@RafaelMunizYT]

3:05 I got this one right by intuition without doing any maths. my thought process was like "if the total mass were 98g it wouldn't make sense because 1g is a small value and there's too much water and too little "other stuff" so that will not even scratch. the subtraction of mass must be something huge to make a difference so maybe half of the initial mass"

[@eduardoroman2691]

La probabilidad de que aparezca un as en la posición nro 52 del mazo es exactamente 1/13, como corresponde a cualquier otro número de las cartas desde el as hasta el rey. En cambio, para que aparezca el ultimo as (o cualquier otro número) en la posición nro 51, la probabilidad ya es algo menor, ya que al 1/13 hay que restarle la probabilidad de que la última carta tambien sea un as. Y así sucesivamente retrocediendo en el mazo, las probabilidades serán decrecientes. Por lo tanto, sí, la posición nro 52 es la más probable para un nùmero de carta determinado, sea este un as, un 2, un 3, un 4, etc etc. Translated for english speakers: The probability of an ace appearing in the 52nd position of the deck is exactly 1/13, as corresponds to any other number of cards from ace to king. However, for the last ace (or any other number) to appear in position 51st, the probability is already somewhat lower, since at 1/13 you have to subtract the probability that the last card is also an ace. And so on by backing into the deck, the odds will be decreasing. Therefore, yes, the position 52nd is most likely for a given card number, be it an ace, a 2, a 3, a 4, etc.

[@DoomRater]

Learning how to use deceptive numbers in video games sounds like something I should pay really close attention to. I could have a lot of fun tricking players with rules that play off of counterintuitive examples. Some like turn based RPGs already do by lying about statistics for most difficulties, but giving the real stated odds on the most hardcore difficulties and suddenly people feel like they're missing a lot more often.

[@wissahdahmastah]

About the aces problem, think of it like this: The chance that any given position is an ace, is equal for any position. However, if the last card IS an ace, it's guaranteed to be the LAST ace. For any other position, there is a chance that it's the second to last, etc. ace. So your best pick is the last card.

[@tonybrowne6737]

You can also imagine counting from the back of the deck and looking for the first ace. It's the exact same problem but the calculation of the probability is much simpler.

[@unclecreepy4185]

I intuitively picked the neighbor with the 10mpg car as saving the most money. Maybe I was influenced by the title of the video. But to confirm, I just picked what happens if they drove 35 miles, which is kind of what the video did after doing the math on 1400 miles. We know the 1st neighbor is using 1 gallon to go 35 miles, saving 0.4 gallons. The 2nd neighbor used 3.5 gallons to go 35 miles, saving 0.875 gallons.

[@Biditchoun]

For th car question I got completely confused, we always measure consumption with L/km here, not km/L.

[@NidraxGaming]

Yeah Or rather L/100Km to be precise. SI master race.

[@diabl2master]

That fact should have been a massive clue.

[@joeshmoe7967]

In single units L/km nowhere as easy to deal with vs km/L. My motorcycle gets 14-18km/L. I multiply how many litres I just put in and poof, I know when I need to stop depending on how I ride. Hoe is that not simple? On the other hand knowing I will use 1/14(.0714) or 1/18(.0556) L each Km I ride does not offer an instant visualization of when my next stop will be. It really come down to what one is used to. I don't think in terms of over all better. I think in terms of better for me personally, as it is my brain analyzing the situation. Km/L still rules!!!! LOL - Cheers

[@kennethlamb668]

That's why in Europe they don't report fuel efficiency as kilometers per liter, but rather liters per kilometer

[@tomymariscal7433]

It’s cool you also put metric 😃

[@captainsnake8515]

The units don’t matter in any of these problems

[@naxaes7889]

@@captainsnake8515 I'd say it matters in all problems. Problems are just as much logic as communication. That's why maths is considered a language of its own. We could have just used our native language to say "ett plus ett är lika med två" instead of "1+1=2". Standardizing and unifying the language we're using to solve problems is an important part of our technological success. Therefore, the units do matter. Using the standardized unit of measurements (SI) is the correct way to do internationalized math.

[@captainsnake8515]

Naxaes I’m just saying that in the case of this problem units change literally nothing. The answer are all the same. The problems could have completely made up units and the answers would still be the same.

[@sfasfads]

The left cars improved by 40% economy while the right only 25% but the right is buying three times the quantity of fuel in the example. It's like forgetting your school lunch and either one friend will give you almost half of his sandwich or you can have three friends each give you a quarter of a sandwich.

[@bscutajar]

I kind of guessed the MPG problem intuitively. The guy on the left is already spending very little on fuel, so even though the percentage decrease is bigger, it is negligible. Going from $3 to $2 gives a better net saving than 0.5$ to 0.25$.

[@maytricks3969]

The thing about the cable is that a 1 foot increase in radius is very small compared to the original radius of the earth.

[@Mu_Lambda_Theta]

Oh right, that cable-around-the-earth-problem I remember that being a question on a test in school. When my classmates and I exchanged results, the good ones (minority) were scratching their heads because of the counter-intuitive answer and the fact that the majority got a different answer. And the students that were not good at maths were terrified considering the fact that everyone who is good at maths got the same answer, but they didn't. I'm sure the teacher liked hear the chaos unfold, he kind of disliked our class because many did not pay attention. x²-64=0 came up on three consecutive tests with 50% or more not being able to solve it.

[@spaghettiking653]

What did people write as the answer for x²-64=0? x=8, I assume?

[@Mu_Lambda_Theta]

@@spaghettiking653 Yup, a good part of his rant was spent reminding us that we should not forget the negative solution. But if I recall correctly some just left it blank, while others decided to use the quadratic formula (not wrong, but still funny)

[@Subpar1224]

@@Mu_Lambda_Theta wait there are just the two answers 8 and -8 right? Or am I dumb lol?

[@ChrisBattrick]

MauLob x=8, doesn’t that just pop out to people?

[@mata91yt]

@@ChrisBattrick It does and then they move to the next question and that's exactly the problem. Cause there's another answer in x=-8.

[@ApocalypsesDawn]

Could you please do a video on the history of trigonometry. Sure i could read about it, but your voice just butters my bread.

[@ApocalypsesDawn]

Like who came up with Sin/Cos/Tan.

[@kaballfs]

3:04 this can be misleading; the way you phrased it implies that you are removing a lot of watter, but it could be phrased in a way such as this is not obvious, for instance, if you say you're removing 1% of the watter contribution to mass (or, as you said it, "removin some such as now it accounts for 98% of the mass"), it means you're removing a lot of watter, if you say you're removing 1% of watter, it means you're removing 1% of watter, or 0.99g of watter, thus implying that it did not lose 50% of it's mass, only 1%. I've seen this problem before, and I'm glad you knew how to phrase it correctly so your calculations actually mean what you said.

[@felixroux]

wattter

[@okie9025]

walter

[@johnnyfrankenstein7973]

Your videos are so interesting and insightful, please keep it up🙏💯

[@Dergoh44]

"And let's say its total mass is a 100g" WOW THAT STRAWBERRY IS HUUUUGE

[@medmoded7767]

On the strawberry one if you turn the water to 97.02g it’s gonna account to 98% of the strawberry

[@rinomonkey1859]

Question of curiousity, How would one go about optimizing their guess for finding the third ace?

[@imqwerty5171]

the optimal guess for the third ace would be the second last card

[@Smallpriest]

39th card maybe?

[@DanteKG.]

51st card because the same reasoning goes. You cover a greater number of permutations by picking the last cards

[@rinomonkey1859]

Even though that would mean that the last card would forced to be an ace? With that reasoning optimum strategy to pick the second ace would be to guess the second card out of the pile because to guess the second ace would be the same as guessing the third ace but checking the cards in the opposite direction, im not saying anyone is wrong but it just doesnt 'sound' right even though there is no math to back up my thinking

[@DanteKG.]

@Bob Trenwith i dont believe the same strategy is used for all 4. I just said that the 3rd A should be 51st

[@RobloxKid123]

12:56 The funny thing here is that State C and State D could literally have a job growth of one fricking job each and still have "50% of the nation's job growth" each. State B would have 50,000% of the nation's job growth..... because 1000*100/2 = 50,000 LMAO. I just found this funny.

[@KinglyJason]

“it’s not always the blatant lies we need to look out for. it’s often the misleading truths.” put that on a bumper sticker

[@kingofgrim4761]

That kinda thinking is exactly how I’ve thankfully escaped from the right side of the political compass

[@TheKarantan]

in the first question, you need to mention whether we talk about absolute savings or relative savings. you were obviously talking about absolute savings. if we would say relative then the answer would be the person on the left

[@Aspartem]

Jeah, that's what bothered me as well. Because during the explanation they suddenly added: "Oh, and both did drive 1400 miles", well duh, that totally inverts the result and then I would've immediately said the right one.

[@RoseRodent]

As a Brit, I heartily approve of you slipping in and out of unit. "If the circumference of the earth is ....km then we can lift it .... feet". Yup, that's the UK alright. Don't bat an eyelid at someone approximating a length as "about 1 metre six inches".

[@michaeltorrisi7289]

As a Yank, I find the idea of the metric system appealing, while in practice, it sucks in places. Like a base-10 system where conversions are easy sounds so much nicer than the hodgepodge that we use here. However, describing a person's height feels a lot more precise in Imperial - the difference between 1.5m and 1.7m doesn't feel like much, but that's the difference between a very short person and someone of average height. 21C and 26C don't seem very far apart, but that's a mild day vs a hot one. Maybe it's just growing up with the Imperial system, but for some finer gradients, it feels superior.

[@aerodynamicist4]

Believe it or not, during comparisons like this, there is diminishing returns with increasing numbers. When I used to ride bikes, in the beginning, it was easy to lose tons of time as I was getting faster. Going for a 60 mile ride at 14 mph average was maybe 4 hours 15 minutes. Increasing that to 16mpg average resulted in a ride about 3 hours 45 minutes. A 30 minute faster ride from a 2 mph average speed gain. Later on, when I was able to average in the 20's, a 20 mph average ride is a 3 hours ride for 60 miles. The 4 mpg average jump from 16 to 20 results in a 45 minute faster ride. Remember how 2 mph avg before was a 30 minute difference? Well, the 2 mph avg now is 22, which is about 2 hours 45 minutes. A mere 15 minute decrease from the 20 mph avg speed. If we get into pro category riding, where riders can average 30, a 60 mph ride is a 2 hour ride. A 32 mph avg ride roughly equates to a 1 hour, 50 minute time. A mere 10 minute difference. That is why cyclists are such weight weenies. On a 10 mph climb that is 8 miles, the time savings can be huge, when compared to a 20 mph ride on a flat stretch for 8 miles. It is a compounding effect.

[@logval8019]

I remember seeing one. It was a heart disease meditation advertising that it was 67% more effective than placebos at reducing the risk of a second heart attack. In actuality, the original risk was, let's say, 90%. The placebos made the risk go down to 87%, and the medicine reduced it to 85%. 5% is 1.67 times 0.03, or 3%. Note: I don't remember the exact numbers but you get the idea.

[@nikm8476]

I would say that result is intuitive though. At least for mathematical people.

[@telecorpse1957]

In my country we describe fuel efficiency using liters per 100km, so taking the inverse was the first thing that came into my mind. I didn't expect the choice of units to make the reasoning that much easier and more intuitive.

[@telecorpse1957]

@Bob Trenwith No, I would spell it "litry".

[@telecorpse1957]

@Bob Trenwith Nah, that would mean letters. Like 'a' or 'ł' or something :)

[@doctorlarry2273]

The choice of units does not matter. gals/mile is the equivalent inverse function.

[@telecorpse1957]

@@doctorlarry2273 I know they are equivalent, but one representation is more intuitive to me than the other when it comes to solving that particular problem

[@socrates704]

The rope around the earth was a pleasant surprise for me, that soon became very obvious. As for the last Ace problem, I needed to find the probability distribution for each position. Turns out, it's not that hard. Let: N = total number of cards (N = 52, here) a = number of aces (a = 4, here) X = position of the last Ace Now, we can readily see this constraint: a

[@fermosofiore1336]

Wow

[@jeff594]

It reminds me of Sheldon's dry erase board. But, for me, any math equation with letters has that effect.

[@tonybrowne6737]

Or you could imagine working from the back of the deck and finding the position of the first ace. Position 52: 4/52 = 0.077 Position 51: 48/52 x 4/51 = 0.072 Position 50: = 0.068 Etc.

[@tomheyworth6020]

The ace one is intuitive imo. Each position has a 1 in 13 chance of being an ace, but with the last position that's definitely the last ace, the other positions it's sometimes not the last ace.

[@xzonia1]

At 6:40, what happened to the 6? It was on the left on line 2, but just disappears on line 3 to never be seen again. ???

[@AWest-ns3dl]

With the card game if you choose a spot (1->52) before this is correct. But if you can make your decision while the dealer flips, you discard all the cards including the third ace. Then you have equal permutations for the following ace location.

[@KimMilvang]

A different way of looking at the problem with aces is this. To win you have to pick a a location with an ace, and there must be no aces after your spot. By symmetry all spots have 1/13 chance of being an ace. It is also clear that the more cards there are after your pick the larger the chance it contains one of the remaining 3 aces.

[@mattc3581]

The way I looked at it, just consider if you were looking through the deck from the bottom first, in the reverse scenario you need to pick the best place to guess where the first ace will appear, (that will be the 4th ace from the normal direction). Obviously the first card on the bottom is the most likely place for an ace since all cards after that you have to hit an ace in the right spot and not have hit an ace in the earlier spots. If the first card is the most likely spot for the first ace it is also the most likely spot for the last ace in the other direction.

[@MrNicePotato]

For the final ace problem, I think of it as the following: If the an ace is at the last position, the other cards can take any order. Since the position I choose needs to be the final ace, any card that comes after the position cannot be an ace, which decreases the number of suitable permutations. Therefore, there are the most number of permutations the cards take with the final ace at the last position.

[@rodrigogaleano5145]

Amazing as always.

[@zachstar]

Thanks!

[@DusanPavlicek78]

In the Czech Republic (not sure if it applies across Europe too), we don't use miles per gallon (or kilometers per liter) but liters per 100 kilometers. That representation turns out to be much more convenient when solving the first problem - because it tells you directly how much fuel each to car needs to travel the same distance. Hearing about miles per gallon is always quite confusing to me :D

[@terran854]

My favorite misleading statistics is: say you have a 2% risk of getting hit by a car. Say texting while walking increases that risk by 50%. you now have a 52% chance of getting hit right? nope, it's 3%

[@traviswhite8112]

Intuitively, I would argue that you have to account for all the money the owners of the left car have already saved comparatively. The left car has been able to drive 3.125x as far the car on the right for however long they have owned it.

[@matheuscastello6554]

is it good or bad that a lot of these arent unintuitive to me?

[@hewhomustnotbenamed5912]

I doubt it's intuitive to you. You probably just did the math. But that isn't a problem. If you apply this thinking process to real life then you could get further then most.

[@zachstar]

Does that mean very intuitive problems are counterintuitive to you?

[@matheuscastello6554]

Zach Star i hope not hahaha!! great vid nonetheless of course

[@andyp2000]

In Europe we specify the gas consumption in l/100km... That makes the result intuitive...

[@sivamynthannadesamoorthy9393]

If it's too easy you are doing it wrong -Mathematics-

[@notsoclearsky]

Me in exams

[@simple45679]

Ao no exorcist reference?

[@notsoclearsky]

@@aman-qj5sx that's what she said

[@rohan1002]

If you are reading the question, then you are doing it wrong.

[@kalstonii]

Sivamynthan Nadesamoorthy “common core” 🥴

[@Yupppi]

KZbin algorithm suggested a revisit to this video and I'm now devastated and shocked that I didn't realise I was watching "miles per gallon vs. km/l" the last time. What a god forbidden mess of units.

[@LesliePiper]

I think that there is no permutation between the aces because their "identity"is irrelevant, they should be considered as "just an ace", hope I'm clear LOL 😆 .

[@Fireball248]

He's talking about permutations of the entire deck, in which case the order does matter, as each unique order of aces (and other cards) is equally likely, which increases the probability of a certain event happening. If you have 24 permutations of even functionality identical aces which win, you are 24 times as likely to win than with just 1 winning permutation. It's important to remember to count different cards (in this case) as separate things, even if they are functionally the same. It's like how when you flip two coins, you count HT and TH as separate outcomes, even though they are functionally the same.

[@nHans]

I studied something like this in college a long time ago. What I remember is that if the aces had integer spins, @Leslie Piper would be correct, but because they obviously have half-integer spins, @@Fireball248 is correct. Same with flipped coins. But I could be wrong 🤣

[@irrelevant_noob]

@@Fireball248 but that's an extra factor of 24 that doesn't really matter anyway. Consider you had taken 4 suits of spades from 4 decks of cards, and used those for the 52 cards in the pack. Now there's no need to care for shuffling aces around, they're all spades anyway. And you've missed it with the coin analogy, the coins being in different spots are obviously different outcomes, but then HHT and HHT are basically the same, even if it's H1H3T2 and H3H2T1. When simultaneously flipping 3 coins we don't need to make them identifiable, if we'll just look at the final (LTR) result.

[@Fireball248]

@@irrelevant_noob if you flip three coins simultaneously, you still count HHT and HTH as different outcomes. Consider the probability of getting two heads and one tail. If you count HHT to be fully equivalent to HTH and THH, then you would calculate the probability as 1/4, as there are 4 possible outcomes (namely HHH, HHT, HTT, TTT). In reality of course, the probability is 3/8, as of the 8 possible outcomes (HHH, HHT, HTH, THH, TTH, THT, HTT, TTT), three of them have 2 heads. Doing things at the same time makes absolutely no difference to the permutations, which means you have to treat each coin separately and make sure they're identifiable. The same is true for cards, except in that case there are many more permutations. In your example, the four aces are still different cards, even if they appear to be identical, so there are still 24 different ways for them to be arranged. It's easy to see that if you imagine marking each one differently.

[@irrelevant_noob]

@@Fireball248 yes, but that's not what i said. i SPECIFICALLY said HHT and HHT, only the two coins that end up heads are different ones, which happened to fall in that order left-to-right.

[@BR-lx7py]

The 4th ace is the same as the 1st ace if you look at the deck from the other end. So you immediately see that the odds of winning is 1/13. The odds of the second card being the first ace (the second to last being the last ace) is 12/13 (first card not an ace) times 4/51. 4/51 is only slight more than 1/13, but you multiply it by 12/13 which lowers your odds much more than what 4/51 vs 4/52 increased it.

[@cheydinal5401]

The chance of the 4th ace coming at the last card is simply the chance that the last card is an ace, at all, or 1/13, or 7.69%. Meanwhile the chance that it comes as the 4th card, is 4/52x3/51x2/50x1/49=0.000369%

[@Rumpael]

Why 1/13? It should be 1/52? Or am I doing something wrong?

[@cheydinal5401]

@@Rumpael Well, it would be a 1/52 chance that for example the heart ace comes last, but 4x1/52=1/13 for all 4 aces added up

[@Rumpael]

@@cheydinal5401 ahhh you're right, I forgot about that.

[@ZER0--]

I've recently had this guy in my feed. Good stuff. Can't believe the cable around the earth thing.

[@scoutskylar]

Here's an interactive graph of the "find the ace" problem: www.desmos.com/calculator/3t8ftm2tmg

[@omerd602]

Yay, another Desmos user! I like the visualization - I just made some additions to your graph that let you vary which ace you want to find (like last ace, 3rd ace, etc) at www.desmos.com/calculator/lfkqqpq8uw

[@chrisjett6232]

One of my favorite counterintuitive math examples is from the 2015-16 NBA season, where for most of the season, the Warriors had the best three-point shooting percentage AND the best two-point shooting percentage, but it was the Spurs who had the best overall shooting percentage

[@mattc3581]

Haha nice. Basically the same thing but I once had to try and explain to a sales exec why, when we had all the customers split into two demographics, and our market share in both demographics was forecast going up next year but our overall market share was forecast going down. He wasn't having it.

[@adityachk2002]

Got to learn new thing

[@georhodiumgeo9827]

Dang you really had me on the radius one. I literally just finished coding some stuff working with rings and circles and thought for sure it would be very small. Great example.

[@onebronx]

Even more fascinating that the clearance between the surface and the cable does not depend on a radius, so it will be the same for the Earth, a soccer ball and an orange.

[@georhodiumgeo9827]

onebronx I did not know that either. I guess C=2 Pi r and I have some catching up do to. Thanks for the tidbit!

[@tighegilmore9202]

A lot of these are unintuitive but, to even an average high school student, are solvable. The strawberry question (or as I've seen it before, the watermelon question), is one that I've asked students to work through before because of how simple the answer is once its worked out but it will wring your brain out to get there if you don't understand it immediately. Understanding unintuitive data is a brilliant exercise and your examples at the end summarise it beautifully.

[@llSuperSnivyll]

When someone says "numbers don't lie", I like to respond with "but they don't say everything".

[@n00dle_king]

The odds of getting the final ace in the 52nd spot is one in 13 as each card type is equally likely to appear.

[@alejrandom6592]

I was expecting the probability of the card guessing game to increase near the last 1/4 of the deck (cuz assuming an even distribution it's most likely there) but I didn't expect the last one to be the best

[@stevejeffryes5086]

Keep your units consistent. Do not give a length increase in feet and compare it to an overall length in kilometers. That is just sloppy.

[@philipberthiaume2314]

If you use metric conversion you can see the difference immediately in benefit to the vehicle user on the right. I live in a metric country and to me, the first 'problem' was very obvious to a point where I was looking for a catch as to why I could be wrong.

[@Juhamakiviita2.0]

10:54 win% on choosing 1st: 0/6 2nd: 2/6 3rd: 4/6 so by picking 4th u would win 6/6 times, but thats cheating

[@PunnamarajVinayakTejas]

For the first question, I would argue that your explanation is incomplete. The mileage being in the denominator isn't what invalidates the percentages. The left car does indeed have a larger percentage reduction in its fuel consumption than the right car (28.57% to 20%). However, since the right car guzzled so much fuel to start with, the absolute reduction in fuel consumption is larger even though the percentage is lower.

[@sujaypoudel4694]

Why is Mathematics always so interesting? Is it because of Zach Star?

[@Schaex1]

It's because of mathematics

[@rapzeh4]

The gas saving problem is actually very easy to figure out if only you presented the fuel consumption in a better way. And what is better than mpg (or km/liter)? It's gallons/100 miles, or liters/100km. To calculate that, you just divide 100 by the mpg. For this example, I'll use the metric system. The car on the bottom left is 100:10.6 = 9.43 L/100km Upper left is 100:14.9 = 6.71 L/100km Bottom right is 100:3.4 = 29.41 L/100km Top right is 100:4.25 = 23.53 L/100km Looking it like this, it's quite obvious who saved more fuel. All you have to do now is look at the difference in L/100km. The guy on the left saves 2.72 L/100km while the guy on the right saves 5.88 L/100km. So basically, showing the fuel consumption of cars as "distance/unit of fuel" is much worse than showing it as "units of fuel/set distance".

[@Posby95]

But the guy on the left saves 28.72%, while the guy on the right saves 18.97%. Percentually, the left one saves more.

[@ABaumstumpf]

Stating fuel"economy" is just making it needlessly unintuitive and errorprone. There is a reason why the vast majority uses the actual useful metric of fuelconsumption per distance traveled. Cause in general you have fixed distances and want to compare how much fuel you need for that distance. And given in this normal way it becomes quit obvious: Car A goes from 9.4L/100km to 6.7L/100km, and car B from 29.4 to 23.5 - also all but the upgraded car A are horrible.

[@nidurnevets]

Maybe you could explain what is often called The Monty Hall Game Show problem, if you have never done that.

[@azertyuiop432]

yep, mathematics are great

[@timmartindale75]

I have an interesting example similar to the power cable thing: In Australia, it's popular for retired people to buy a camper or caravan (US: mobile home or trailer) and drive around the entire continent, usually in an anticlockwise direction (US: counter-clockwise). These people are nicknamed "grey (US: gray) nomads" and many will tell you that you should swap your tyres (US: tires) right with left and left with right because over such a long distance one side (the one farther from the centre (US: center) of the circuit will do many more kilometres (US: miles). I once told one of these grey nomads that it only travels about 9 1/2 metres further but no amount of demonstrating the maths (US: math) or logical argument could convince him.

[@Sasoridellasabbia]

The fact that you're doing physics with imperial units saddens me.

[@Kenfren]

I think you mean freedom units

[@nocakewalk]

For the cards, every spot in the deck has the same probability of being any of the values ace->king, I.e. 1/13. For the last card, all of the 1/13 for ace corresponds to it being the last ace, while for every other position in the deck, the 1/13 is shared between the first, second, third, last in some way.

[@jimmysyar889]

first at 3:00 AM lol

[@rysea9855]

Wait what.. How? The video came out only 3 hours ago

[@jimmysyar889]

Temmie Village Zach is a super chill dude so I Patreon’d him and he sends the links early to us among other things. It’s only a dollar a month which is only $12 a year and it really helps him out a lot.

[@henrique.c72]

Great stuff Zach! Thanks for that video.

[@absoluteexistence8279]

Never stop this mathematics series

[@paulcook2961]

In my head, I rephrased the card question like this: since every position has exactly a 1/13 chance of being an ace, pick the position that is least likely to have an ace later than it.

[@egilsandnes9637]

My favorite counterintuitive mathematical/statistical phenomenas are the Monty Hall problem and Simpsons paradox.

[@Simpson17866]

@@jamesedward9306 Possibility A) The guest guesses right the first time. HOST: I've written down a number between 1 and 1 million. (it's 783,689) Can you guess what it is? GUEST: 783,689? HOST: Before I confirm or deny your guess, I can tell you that the number I wrote down is EITHER the number you just gave me OR 700,000. Would you like to switch? Possibility B) The guest guesses wrong the first time. HOST: I've written down a number between 1 and 1 million. (it's 783,689) Can you guess what it is? GUEST: 700,000? HOST: Before I confirm or deny your guess, I can tell you that the number I wrote down is EITHER the number you just gave me OR 783,689. Would you like to switch? In both cases, the second round has to end with the guest choosing between one right answer and one wrong answer, and the host has to use their secret information to make this happen. If the guest guessed right the first time, then the host has to use their secret information to offer one of the many wrong answers. If the guest guessed wrong the first time, then the host has to use their secret information to offer the only right answer.

[@zachstar]

Love both of those

[@xj-vn4eo]

@@jamesedward9306 Yes it is quite mind-boggling. This is how I think of the problem. Thing about Monty Hall Problem is the additional information on the unchosen options. The probability on the chosen one does not change, which reflects the amount of information we know about the thing we choose. However, as we know more about the unchosen ones, the probability of the unchosen ones now changes.

[@xj-vn4eo]

Monty Hall problem and Simpsons paradox are my favorites too. I encountered both of them when I was doing research.

[@shadowpenguin3482]

Regarding the casino riddle - if you ask the question what number has any ace, any number you give has the same odds. By picking the last card you maximize the odds that actually matter

[@danyakloppers9288]

But why do you add the 6 to both sides of the equation?

[@Thundermikeee]

The percentage thing reminds me of an online argument i have seen recently concerning the pandemic: Person A : it is pretty bad though, 1 in 50 people die from it. Person B : You are lying, they say there is a 98% survival rate. which shows again, that humans really really really suck at interpreting probabilities other than 50/50 on an intuitive level.

[@Posby95]

The framing effect...