Fair Dice (Part 1) - Numberphile
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- Опубліковано 4 лип 2024
- Probability expert Professor Persi Diaconis (Stanford University) talking about dice.
More links & stuff in full description below ↓↓↓
Part 2: • Fair Dice (Part 2) - N...
Tadashi and Dice: • The Most Powerful Dice...
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Diaconis/Keller paper on fair dice: bit.ly/FairDicePaper
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"There are 5 fair dice."
*angry d2 noises
*d2 lands on its side
deniz-usta Gedik *angry d20 noises.*
@@The_Murder_Party *Angrily rolls percentile dice*
The Procastinators I mean to be fair percentile are two d10s, but this is fair.
Never had a D2. Is there a die for that? Or is it a coin?
I love this guy's name. It's like the name someone would have in a medieval fantasy story.
"Quick, m'lord! We must reach the king's statistician Persi Diaconis before sundown or all hope is lost! He's the only one who knows how to make a fair die out of non-regular polygons!"
Finally, a name for my NPC wizard.
It is rumoured that the legendary artificer, Persi Diaconis of the Order of the Logician, once created an object that appeared to be a normal dice. In truth, this dice was filled with magic, and power, and quite a lot of hatred, and its roll would influence the very fate of the world...
Yeah his name is awesome! :D
Be careful not to attack him though, he does 2d10 math damage.
It sounds like something out of the game twisted wonderland and I love it
All a D&D player wants to know is whether the D20 is a fair dice. ;D
That's the regular icosahedron, one of platonic shapes which he described as fairest.
Yes. I did wonder if the d20 was in the standard platonic solids set... I suppose the most questionable die used in D&D (aside from one for which no dice formally exist, such as d100, or d2) would be the d10.
All the others are platonic solids. d4, d6, d8, d12, and d20, and then d10...
i guess
It never is...
Wil Wheaton might disagree.
Fair dice: Not the DM's
As a Dungeon master I approve this message.
Your channel turned me from a person who thought they hated maths, to someone who appreciates its beauty, thanks!
I've always appreciated math. Its basically our way of explaining the universe.
I'm just absolutely garbage at it, and that makes me bitter.
@@Guru_1092 I know just how you feel. I share that bitterness as well.
In high school our physics teacher used to choose people for oral exams by throwing a 30 sided die lol
that sounds so weird and disgusting...
redbeam_ why?
@Nate his mind is in the gutter, "oral exams"
Emerson Harris I know. Im just saying that it shouldn't sound dirty and that his mind is in the gutter.
is it fair?
For anyone interested, here are the names of the shapes shown at 7:20
Left to right, then top to bottom;
cube/hexahedron, octahedron, pentagonal hexecontahedron, pentagonal icosahedron.
*opens gaming shop called "Die, die, die!"*
The missing one is pentakis dodecahedron
@AINIEL YABUT the "not sure" shape is actually a rhombic dodecahedron
You're a legend. This comment should be pinned.
The way this man describes dice reinforces the idea that there's a very fine line between insanity and genius.
I once saw 7-sided dice, that were basically extruded pentagons. And my initial reaction was that there's no way such a die could be fair, but then I thought about it for a few minutes.
If you have a pentagon that's extruded very thinly, like a wafer, then it'll be biased in favor of the two pentagonal faces and the other 5 faces will hardly ever show up. If it's extruded several feet, then the two pentagonal faces will hardly ever show up and you'll usually get one of the 5 others. So there MUST be a sweet spot in the middle where the biases cancel out, and you'll get one of the pentagonal faces 2 out of 7 times!
Ah, I see now he covered that in Part 2! Maybe not so fair after all...
meh, tops are really useful for X sided dice, take the dreidel for example.
INFINET SIDE DIAS!
It'd just make more sense to have a heptagonal prism and round the edges for a 7-sided die. That's what they did for the 3-sided die.
I think standupmaths (a side channel for Matt Parker) did a video on a similar problem, finding the dimensions that would make a cylinder work as a 3-sided die. I remember that when they were trying to calculate it, they got two different results based on what area was selected for the random distribution, and they tried rolling a bunch of such dice themselves, but I don’t remember if they got any results from that. There were a bunch of people in the comments (including me) who were saying that it would never work due to the fundamental lack of symmetry between the sides of the cylinder. In particular, the way you throw it majorly affects the results; ie, throwing it so it rolls along its axis means that you would get far fewer of the two ends and too many of the band.
Essentially, isohedral dice are fair because all the sides are interchangeable. However you throw the die, it can be oriented beforehand so that it has the same overall shape and thus rolls the same way, but any other side you want ends up on top. Because of this, if the original position of the die is random and unknown, so is the resulting roll. This is not true for the cylindrical die; you can’t put the band in the place of one of the ends, or vice versa. So it can never truly be fair. Just use a top for stuff like that, or try cubical dice and count opposite sides the same.
PS: speaking of the starting position being unknown, some people there tried to argue that this argument was wrong because ordinary dice can also be affected by the way you throw them. For example, some people have become well-known for cheating at craps via something called the blanket roll; basically, they throw the dice so that they roll around only one axis (similar to what I said about the cylindrical die), and like that, the two sides at the end of the axis (in this case, usually 1 and 6) were less likely to end up on top. My answer to that is that this effect depends on the original position of the die being known; the blanket roller has to look at the dice and put the 1 and 6 where they want them for this to work. If they just grabbed the die and threw it without looking, their chances of getting, say, a 1 wouldn’t be any different from before. Again, this is because the dice’s sides are interchangeable; the cylindrical die’s aren’t, so they can’t be thrown fairly.
"Small changes in the initial conditions change what side faces up"
In othere words, dice are not just random, but chaotic :D
Is it possible to make strategically unfair dice?
I've always wanted to make a 12 sided dice, with the same probabilities as two 6 sided die
Maybe you could make the faces with the central values larger than the ones with the extreme values? Make 7 huge and 1 and 12 tiny.
one problem with that is that two 6-sided dice will never land on a sum total of 1.
There are only eleven possible outcomes to a 2D6 throw. ;)
Still an interesting question, though.
jmiquelmb: I would say the opposite, because big faces are more stable, thus, if 7 is on a small face opposite to 12, probability would be correct.
Josselin Luneau If I undertand well, larger faces would mean more probability. Then, 7 should be in the larger one because it's the most probable result, while 2 and 12 (not 1 and 12 like I said incorrectly before) should be on the smaller ones. It's easier to get a central number on a two dice throw because there's more possible combination outcomes (7: 6+1, 2+5, 3+4, 4+3, 5+2, 1+6; 2 and 12: just 1+1 and 6+6)
6:47 I was waiting for him to mention a d10, then he invented it
What he described is not a traditional d10, a d10 is a dodecahedron with two faces extruded to points. On a d10, the faces interlock with two opposite faces. His d10 only has one opposite face connected to each.
There is one of the interests of simulating chance: once they're balanced, all virtual die are fair.
"Talking to me about dice and fairness is like talking to a California wine person about wine - it can go forever."
Please do!!
That d4 awakened a deep anger within me
4:20 reminds me how a standard 3x3x3 Rubik's cube works. When you make one full turn on one side it stays in the cube shape. But when you change the shape into lets say a Rhombohedron while still keeping the same turning cuts as a standard 3x3x3 Rubik's cube it starts to change shape when mixing it up.
So I tried making an actual list of the fair dice as shown at 7:23, using these visuals and the original paper. Here's what I've got:
D6 - regular cube
D8 - regular octahedron
D60 - pentagonal hexecontahedron
D24 - pentagonal icositetahedron
D60 - pentakis dodecahedron
D12 - rhombic dodecahedron
D30 - rhombic triacontahedron
D24 - triakis octahedron
D4 - regular tetrahedron
D24 - tetrakis hexahedron
D60 - triakis icosahedron
D60 - deltoidal hexecontahedron
D12 - triakis tetrahedron
D24 - deltoidal icositetrahedron
The infinite family of bipyramids (pictured is the triagonal bipyramid I believe)
D48 - disdakys dodecahedron
D120 - disdakys triacontahedron
D12 - regular dodecahedron
D20 - regular icosahedron
After that I am kinda lost. The visuals are confusing me a bit, cause the deltoidal icositetrahedron and disdakys dodecahedron seem to be there twice (at positions 14 and 22 and positions 16 and 21 respectovely). The second to last shape also looks like just a regular octahedron, which is already listed before. The last one also looks like a rhombic dodecahedron, also already listed. Furthermore, after reading the original paper, I've come to understand that the fair dice are: 5 Platonic Solids, 13 duals of Archimedean Solids (known as Catalan Solids) and 2 infinite families. Based on the paper I think the infinite families are supposed to be bipyramids and trapezohedra.
But that's all I got and that's just 20 families. The video says there should be 30, but I can't figure out what the remaining solids in the video are supposed to represent and the paper seems to be talking about only 20 families as well, unless I missed something. If anyone has any ideas, please let me know.
My man that was a lot of work
"I have a thirty sided dice" Who wants to bet that he got it to play D&D
What TTRPG needs a d30?
I will.
Just bought one for my son to use as a DM.
"There are only five fair dice, d4, d6, d8, d12, and d20" *sweats in white wolf*
D10 bruh
Wild Magic Sorcerer: *Sweats in D100*
D10
@@gillasosaurus d10 isn't fair. Vertices of 4 or 5 depending on position.
@@AlexH274 k, but they're statistically equally likely, which is what matters when it comes to fairness
"(...)new Tadashi video soon, that's something to get excited"
Oh Brady, you know your audience so well...
This was as mathematically and philosophically as beautiful a video ad any other numberphile video as I've watched ever.
Dammit! I thought this video had the man with a thousand Klein bottles when I saw the thumbhnail but it was an impostor.
Right?!?
Ah, but this is Perci Diaconis, a magician who studied with Dai Vernon. Just as interesting. Problem is, Brady has no reason to ask him about it.
Cliff Stohlen Identity
Make it stop.
6:45 did he just say "fivegon"??? 😞
'Fair dice' feels like it should be a saying ...
I think what you said is fair dice
jmiquelmb
haha! yeah, just like that :)
I cannot talk about probability all day like Prof Persi, but I'll watch any Numberphile video, so- fair dice.
I think what he said is but a parker square of a fair dice
No dice.
Casinos HATE This Man: Find Out Why
Yeah, right, because Casinos make so much less money before him than after him. What the heck are you talking about? I'm not sure you know how Casinos work. Either that, or you haven't thought the statement through. The point where the rubber meets the road is the point where the rubber meets the road, and I can prove that mathematically.
love this! being a gamer i roll dice all the time, so this is a great video.
Heard him speak in New York a few weeks ago. Really entertaining and insightful speaker.
I love how often I come up with an issue with something someone is saying, and the question is brought up in the video.
If all the dimples in a golf ball could be numbered, would it be fair??
Only if it were possible to space every dimple evenly apart, and make sure all dimples have the same number of adjacent dimples in a way that is identical and symmetrical to every other dimple. Someone who know more about the design of golf balls could probably say for sure.
EDIT: After watching a few videos the best I can really say is; maybe? I'm pretty sure golf balls are carefully designed to meet the right criterion for fairness, because a fair dimple placement is required for aerodynamics, which is the whole reason golf balls have dimples in the first place.
Probably not. If you look closely, you see that some dimples are next to five other dimples, and others are next to 6. This means that the symmetry of a golf ball isn't face-transitive.
It could be possible to create a golf ball to be experimentally fair despite this, but why would you want to? Just use a random number generator or something.
Since it's round, it wouldn't stop turning until something stopped it or it finished all the momentum, so it wouldn't land and you wouldn't be able to recognize the face it shows
There are 100 sided dice, they look like golf-balls....
Yes, if it were perfectly spherical and dimpled in such a way where one is directly across from another through the center.
Great to see Game Science dice in use.
Zocchi dice!
I'm guessing he chose them because of their purported fairness.
I got my game science dice to overcome dice superstition, and have since become superstitious about using any dice that aren't my game science dice
Adderkleet that’s what I thought! I saw the clipped corners on the d4.
I was hoping he would said something about the sprue discoloration.
this brings back memories of crystallography. Good o'l mineral symmetry groups.
Wasn't anyone else blown away when he said a guy roled a dice 3,500,000 times and recorded the results?!?!
what about this dice guy? did he end up with equal results for all 6 sides of a die?
don't think so. they mentioned the weight difference, especially between 1 and 6, shortly after talking baou him.
i guess he had less 6s than 1s.
he would of been infinitely close.
Pretty sure it would be more 6's. 1 is less material removed therefore heavier and more likely to be on the bottom. The inverse is true for 6.
I think you would need an absurd amount of data to notice that difference to be honest.
Well, he rolled it 3.5 million times, so...
Pro-fair-sor Die-cone-is...
:)
I'll never forget his name again!
I took discrete probability at SUNY Albany with Professor Martin Hildebrand, whom I think had this Professor Diaconis as his PhD advisor. I imagine this professor has advised many PhD candidates, but Hildebrand seemed to stand out as pretty brilliant (Harvard PhD after all). Any of you guys take classes with Professor Diaconis or any of his "descendants"? I do believe my prof at UAlbany has (obviously) published with Prof Diaconis as well....
Nomen est omen!
Ahhh, procrastination brings me back to my work, AND to E.T. Jaynes's book I'm in the process of reading!
Hahahaha! "Caught me there... Let me AMEND the statement of my theorem...." - Love it...
So, the d30 is somewhat similar to a soccer ball, in that it's a shape made of a pattern of two different subshapes, namely 3x5-rhomboids and 5x3-rhomboids, and because of that, some values are more likely to show than others.
Now I'm really curious as to the frequency distribution of a d30. If anyone knows where to find that, please drop me a line!
No, the d30 is a rhombic triacontahedron, it's made of one shape - the golden rhombus - and all values are equally likely to appear, assuming the weight is evenly distributed.
@@RDSk0 Yes, it's all one facet shape. No, they're not all as likely to show up because of how they're grouped.
6:00 The edges on a rhombic triacontahedron ARE transitive. See Wiki on the triacontahedron
Ooooh, he got me all excited at the end there about the next Tadashi video!
You're more than welcome Brady! Best channel ever!
"same specific gravity"... Oh boy.
*Several D&D players including myself are typing*
Thanks for making great videos with smart people Brady! :)
For an algorythm/symetry nerd like me, this is as entertaining as watching a movie in a cinema for normal people
Would a tesseract be a fair die though ? =D
A hypercube, yes.
Yeah, it would have the same chance to land on all 8 volumes.
They have made a video called "Perfect Shapes in Higher Dimensions" that is kind of this problem in higher dimensions.
+31nar288
Correction, all 8 _cubes_.
(Nobody says that a 3D die lands on one of its _areas_.)
Liked anyway.
Cheers
;)
[Edited to add] On second thoughts. maybe even _cubes_ isn't right. What (3D) corresponds to 2D _sides_?
You'd have to make sure you were throwing it in 4 dimensional space too, though, since in our 3D space it would only flip in 3 different directions which would make it unfair.
is this the klein bottle guys brother?
Ummm.... no.
Well you username checks out. I trust you.
Persi Diaconis.
Cliff Stoll.
Maybe half brothers...
It's the same guy, people.
Geez, I hope you're all joking...
Very interesting! I had the same idea when he showed a dice made from a pentagon.
"turning it around three things"
I love this guy~
Actually, all the edges on a rhombic triacontahedron (the 30-sider) are in fact the same. You can map any edge onto any other edge the same way you can for the faces.
The fair dice shapes are sometimes called "Catalan Solids" or "Archimedean Duals".
I'm 5 years late to warning you not to argue with the old mathematician.
@@DanielFerreira-ez8qd I'm not arguing. I'm just stating a fact. And why in the world should anyone need to be "warned" not to argue with a mathematician?
@@PhilBagels I didn't mean "argue" in the aggressive manner, just that you corrected the math man, which is a humorous thing to do in a scenario where you could be corrected immediately. This ain't one of those obviously, I'm just messing around
@@DanielFerreira-ez8qd But I'm right,
Rolls damage: 40k6
Thanks for legibly showing the complete paper!
You know he is a genius as he closes his eyes while explaining because he is visualizing it.
7:20 That audio editing though...
I guess the sphere is the fairest of them all, but then again it can sit in the middle of multiple answers
Make it transparent and have a multi-directional laser in the centre that fires out vertically so you can better see what number it's landed on. Cheap and perfectly safe. Problem solved.
In my opinion its not because the way it rolls. The sphere if you look in terms of phisycs it only rolls at 1 direction, by the other way cubes turns on all directions. I don't know, i'm just giving my thoughts
now that i think about it , a sphere has only one side...
Of course, it depends on the rolling substrate. If it's lumpy, the sphere is super fair (if you've just labeled sections as sides), but as Jorge said, you completely control the one axis of rotation if you roll it on something flat.
Always a critical failure.
Absolutely fascinating - I'd love to see a big venn diagram of face transitive, edge transitive and vertex transitive solids (maybe I should get to work on it myself...)
I saw a 7-sided die. It was a pentagonal prism, so two sides were pentagons and there were 5 sides that were connecting them. The guy who made it was talking about how if you look at it, the pentagons look so much larger than the smaller connecting sides, and you'd think there was a higher chance of the pentagons landing up, but there actually wasn't. He made bets with people where if it landed on a pentagon, he'd give them one dollar, and if it landed on the others, they'd give him two. Of course, he ended up winning because it was a fair die.
Спасибо за видео очень интересно и полезно
4:45 as you can see, there is a small triangle at each corner of the tetrahedron, which means there is a chance, an astronomically small chance, that the die can land exactly on the small triangle.
Exactly. It's just an unfair 8 sided die.
This video is insightful and delightful and I feel smarter having watched it.
never thought i would be this engaged in a video about dice.
A fair dice may exist, but a fair throw does not. You could make it very close to a fair throw, but conceptually it's impossible I think.
A fair throw is where you didn't intentionally manipulate the chances. The matter of the dice throwing to get a result from a fix pool that you didn't know before the throw.
But, it's needed to make it "unfair" to make it possible to determine a value. In a theoretically perfect world a theoretically "perfect" throw would cause the dice infinitely rolling without stopping, because that perfect you throwed. If not, and we assume that the dice is not mathematically perfect object, it would stand on an edge, like a coin. Throw a coin. Heads or tails, but you throwed so perfectly it didn't lean to either side.
The long story short is that as long as the individual panels of the polyhedrons are made of "equilateral" shapes (All sides AND angles are the same) then the die should be fair in terms of symmetry.
What a fantastic abstraction
Spheres are fair too. Infinite sides. Equally chance on any side facing up. Also, any cylinder based dice that you roll are fair.
Give the sphere dice (a d^3 if you will) a UV texture or color ramp, that way you can extract an X and Y dice value out of every roll.
This is how Cliff Stoll would look and act like if he wasn't constantly high
yeah
0:08 "Tetrahedron" That ones got 8 faces. I had a dice with rounded edges, and it managed to stop on an edge. Only once though.
As an RPG dice goblin, words cannot express how utterly stressed out 2:30 made me
What about a case for the three sided die... you can see three sided dice shapes within the old pieces for risk which symbolized 10 men. The 1 man pieces were squares, and the 10 man pieces aka cannons were the shape of what seemed a fair three sided dice.
I have a D120 I use for D&D random tables, I wonder if that's considered fair
i dont think so. do you use the official PHB? cuz if you do there are only d100 random tables and to trow a fair d100 just use two d10.
Well you could determine a fair number in 1-120 with 2d10, 1d20 and 1d4.
Roll the d4:
at [1,2,3] it's up to 1-100
at [4] its 101-120
In case of [1,2,3] simply roll 2d10 to get the digits for a d100
_or_
In case of [4] just roll the d20 for your 101-120
Is there a beholder listed on the table? ... that's not fair!
Have an awesome day everyone! :)
no
I am in pain
+NonTwinBrothers im also sick....:(
you to buddy :D
Don't tell me what to do.
Excellent video
Thanks for the video!
Rohan is really suspicious of dice now
Goddamit, is this a jojo reference?
@@lilysdong1457 xD yes, yes it is, and I hate how it got even here
Gamescience dice! They're the best.
I thought i recognized those immediately, shame I still don't own a pair.
First thought that popped to mind when the video started. =)
Fun bit! The dice they are tossing in the examples, the ones with no paint in the numbers and very sharp edges? Those are from Gamescience, a company that prides itself on making them that way because the rounding of the edges can cause flaws with their balance and make them less random.
The gamescience dice made the video so much better
The casino dice are pretty
I want some.
Maybe I’ll get some on amazon tomorrow.
If you worry about the fairness of dice, you are either a math professor or a munchkin.
Or a D&D nerd.
There's a clever trick to ensure you are flipping a fair coin (the logic can be generalised to dice).
The idea is that you start with a H/T coin with roughly 50-50 ratio for landing: and by 'roughly', I mean that 0.01
Everyone: Goes nuts because of not-cube dice.
Dungeons and Dragons players: First time?
How do you roll a die 3500000 times?
You build a machine to do it, with maybe 20 or 50 dice being tossed at a time, incorporating a camera to record them after each toss, and recognition software to count each outcome from each die, at each toss.
At the beginning of Part 2 he shows how a student did this as a project; he used 12 dice at a time.
Or you vedge out with dice while watching TV for 15 years.
how long have you been on youtube
You roll it once, then you wait until it comes to rest, then you pick it up and do it again. Do that over and over until you reach the number you are after. I hope this helps.
Your watch as much content as a typical 12-year-old and spend that time rolling, rolling, rolling.
Rolling a dice every 3 seconds. No resting. No sleeping. That's about a year.
old news to every dungeons and dragons player lol
you can create fair dice that have symmetrical edges, vertices and sides IF you allow the faces to have even a tiny amount of curvature. For example a toblerone that doesn't have flat ends, but points.
to extrapolate the theory, BLOWING on the dice for luck, could influence the outcome of the dice roll !
one side of the die would be warmer than the other, influencing the dynamics of the roll ... great video
SO HOW CAN I WIN A THE CASINO
Don't play.
own a casino
@@Xormac2
That also works.
You can start by proofreading your comments before posting, so that you don't come off sounding like an ignorant moron. Morons don't fare too well in casinos. Next, go to videos by 'Dangerous Arm Craps' and watch and listen. Then put it all, everything you've got, across the numbers as soon as you get the dice and don't work them on the Come Out roll. Hit 4 numbers and pull the bets down. Send what you came with home and play only with the profit. You're welcome.
The best way to win the game is to not play
why am I still watching this at 1 am?
I bought a set of 'Rogue' dice where the d6 were in a poison bottle design. I quickly discovered that I could better control the outcome by how I released them from my hands when rolling.
That's the most rouge thing i can think of.
I know about octahedrons from the Octahedron of Transcendence (Timotainment)
It really bugs me that his 4 sided die have the corners shaved off !
Have you ever tired rolling a tetrahedron. The corners are very sharp abd make jt hard to pick up. Shavjng them off makings the die more weildy.
lots of time i'm an avid pen and paper player, playing pathfinder mostly now
That's to keep it from hurting when you step on it. Dice roll off the table more often than you think.
only five in the 3rd dimension!
I'm interested in what types of polychoron dice would be fair, assuming a 4-dimensional space.
The same ones. and the hyper diamond.
There's two more (the triangular and pentagonal bipyramid) plus several other if you allow using non-regular faces.
@@robo3007
Regulars only, sorry.
There's only three in 4d.
One, Infinity, five, three, three, three...
I want a d1: a d6 labeled 0,0,0,1,1,&1. Roll it with another, then a d4 becomes a d5; a d6 becomes a d7; etc., all fair rolls.
dude, i will allow this guy to go on forever. For once, i feel like i understood the whole video.
it looks weird that his eyes seem constantly closed
old people HAHAHA
he's trying to image the geometrical shapes he's talking about, makes stuff easier for some people ;)
Just like Brock.
At the start, those 5 shapes would be called Platonian solids, technically speaking.
Yes, but it's, "Platonic."
ffggddss Close enough.
I just thought of a way of finding the "most" fair dice of any number. You must start with a perfect sphere and cut it the minimal number of times to have the desired number of sides, and so that each side is equal in shape & size, whilst retaining as much volume of the original sphere as possible.
For gaming, if not geometry, you can have fair dice of ANY value. Make them in the shape of a prism with that many sides, and sharpen the ends to points or round them off, so the die won't land on an end and stop there. To help you visualize it, a regular everyday pencil, if not cylindrical, is a six-sided prism. Sharpen both ends and you've got a prismatic d6. And obviously these can be made with ANY number of sides greater than 2.
D2 = a coin
Wrong.
lol no
Any nerd could have told you what those original five dice are:
D4, D6, D8, D12, and D20
I have a d24 and a d60 and a d30 along with the standard polyhedral dice
Fun fact: when you flip a coin there is actually a greater chance that you will get a streak of 9 heads or nine tails in a row than there will be that you never encounter that length of streak. In fact one professor asked students to flip coins and record the results. The number of flips was pretty large, around 300 and some students got lazy and made up the results. However the professor could tell which data sets were real and which were made up by looking at if the data contained streaks of 7 or more. The fake results that students made tended to alternate frequently and not include very many streaks of heads or tails in a row.