# How random is a coin toss? – Numberphile

There it is, that’s hard to get. I don’t know if your camera is fast enough to catch that. I’ll do it until you cry “Uncle”.

There you go. So for many people — me in particular, viscerally — this is as close to a random phenomena as I know. And yet you can think about it, if you knew when the coin leaves your thumb ( faint ringing sound ) if you knew how fast it was traveling, and if you knew how many times a second it was turning over Newton tells you, if you know the speed,

how long it takes you to come down and that if you know how fast it’s turning, you know whether it will come up heads or tails. Coin tossing is a deterministic process. So what’s random about it?

But first of all, let’s try doing a little of the math of tossing a coin. I’m going to say feet per second,

I know that’s unpopular. And here’s revolutions per second. So a dot on this plane is a coin toss. A dot on this plane is how fast it’s going,

and how many times a second is it turning. So for example, a dot here near the axis, the speed is high

but the revolutions per second is not so much. So that means it’s going up like a pizza. It’s kind of doing this, and not really turning over. If it’s close enough,

it might be just kind of slowly turning. So it might be that it doesn’t turn over at all. Similarly a dot here, very near the axis up here, is a coin which was thrown

with a lot of revolutions per second but very low speed, so it never gets very far. ( faint ringing sound ) So there’s a region like this, of initial conditions

where the coin never turns over at all. Any place in here the coin never

turns over at all, there’s some region. And we can figure out what this curve is. It’s just, you know, a first course in physics. And then there’s a region

where the coin turns over once. Initial conditions here, the coin just turns over once. And then there’s a third region

where the coin turns over twice. And then there’s a fourth region

where the coin turns over three times, which is the same as turning over once. And these regions alternate. When you do the math what you find is

these regions get closer and closer together, so that small changes in initial conditions make for the difference between heads and tails. And for most of us that’s what makes coin tossing random. Now you can ask, when real people

flip real coins, where are we on this picture? I want to know how fast is the coin going when it leaves my hand? Is it going 5 miles an hour?

Is it going 20 miles an hour? Is it going 40 miles an hour?

It makes a difference. So I got a friend with a stopwatch, and I went: “1, 2, 3, flip!”

“1, 2, 3, flip!” and we got sort of synchronized, and so he could tell how long it took the coin to come down, and from that I could tell how fast it was going. Now, it took three hours to get that data,

if you think about it. To get the data about how fast is the coin turning I had a good research idea which is the following: I took a coin and I took a piece of dental floss, you know the tape type dental floss,

about three feet long, like that, and then I flip the coin up,

and then I could unwind the dental floss to see how many times it had turned

until it was flat again. ( faint ringing sound ) The floss twists around. Now, it’s true that a coin flipping in the air with a piece of dental floss is different than the coin

just flipping in the air unimpeded. So as with any measurement process the measuring interferes with the randomness. But you hope it’s a third order effect. Based on those kinds of experiments and a lot more of that, I know where we are on this picture. What was funny is how fast the coins go.

A typical coin flip lasts, comes up and lasts about half a second. When you translate that,

that means the coin is going up and back about five and a half miles an hour.

That’s how fast this coin is traveling. Now I have to say where we are on this picture. This dot where we are in this picture is about 5, this is 1. And in the units of this picture five and a half miles an hour

comes down to very very close to zero, but the revolutions per second

were 40 units up in the picture. So this picture says nothing about real coins, but the math behind the picture tells you how serrated you are

up there where you can’t see. It seems to be true that coins the way we flip them are fair to 2 decimal places

but not to 3. It’s not perfectly 50/50. In work with Susan Holmes and Richard Montgomery we showed that coins the way real people flip them it’s about .51, so one in a hundred off. And it’s not .51 to “heads or tails”, it’s .51 that it comes up the way it started. That is, if you started heads up

it spends more time heads up. And I don’t care how hard you flip the coin, you can flip it to the moon.

That bias is a provable stable bias. And it’s not a “I think” or “from experiments”,

this is a theorem. When real coins flip, when we get your

high-speed camera, you’ll be able to see that they don’t turn through

their axis of symmetry in that way, through this plane, they precess and they do pretty complicated things. And because of the precession

it’s not just the simple two parameter business. The real analysis of flipping a coin, it’s 12 parameters. Actually describing a moving coin is a 12 dimensional problem.

It’s not something I can draw a picture of. We did the analysis in the 12 dimensional space and put in some data from

real people flipping real coins. We had to measure the distribution of angular momentum when real people flipped real coins. And that was quite challenging, but we did it. So based on all of that analysis, when people, when you flip a coin it’s about .51

to come up the way it started. So it’s better if you see its heads,

it’s better to guess heads. It is also true that if you take a coin

out of your pocket, or jiggle it in your hands,

that probably makes the start random. And then physically flipping a coin

won’t destroy that randomness. It’s not as if the coins have a predilection to kind of favor heads when they come. Brady Haran: This video was made at

the Mathematical Sciences Research Institute. Now you might be wondering,

is it more fair to catch the coin

or let it drop to the ground? You also might be wondering,

is it more fair just to spin a coin on a surface rather than tossing it through the air

in the first place? Well Professor Diaconis has discussed both of these with us, and depending on when you’re watching this,

that video is coming very soon. Or, maybe it’s already uploaded.

So check those out, and thanks for watching.

P00Rn

1% extra at saving oneself from Anton Chigurh.

Isn't there a possibility that the coin could land on its edge? So neither heads or tails? How come he does not address this possibility?

He missed the fact that once in a while it lands on its edge. About one in 5,000 times.

Why do people love to complain about measurement units? It's all arbitrary anyways. If I see a metric unit I don't know i'll just convert it. Why do people go out of their way to complain? I never see comments complaining about videos being in English instead of their language?

The animation shown does not match what the speaker intends to say. Low Speed/High Revolutions should be such that the coin does

notturn over. I.e. the coin only rises about 1 mm.feet/second? miles/hour? you are kidding right.

The dental floss idea was a very clever way of measuring the number of spins.

does the amount of metal (or the picture) on each side of the coin not count for any irregularity? I've always heard that a coin toss is effected by how the coin is printed or forged or whatever the process is that distributes the mass of the coin across the 2 faces

For instance some of the state coins in the US have very little 3D metal on the state side which would make the head side hold more of the mass for the coin

"We did the analysis in the 12 dimensional space"

Something you should not here in a video regarding coin tosses.

Someone could make an app that takes input from a high speed camera and predict what side will come up before it hits the ground.

So when they do the coin flip at a football game, the team captain should always pick the side of the coin they see pointing up. Wow. The New England Patriots have just found a way to gain another edge.

I use the metric system but who cares if he used the imperial system just this once? I truly never saw the problem in using both: they're both different languages used to say the same thing. Its just as silly as asking someone to speak to you in English if they are well versed in Spanish only. Although it is simpler, since it doesn't have different syntax, the semantics and knowledge of both are significantly different. He probably is more comfortable using the imperial system, its your problem that you can't do simple transposition to understand his imperial system of describing quantitative data.

What this video tells me: "we were so bored that we did some calculations to find out useless facts"

But as a Jedi Knight, I will always guarantee that it lands in my favor.

I could totally see what this guy was saying. I believe it.

Holly shit, this guy is a mathematician? He actually takes 7 minutes to explain a simple concept that could be understood under 1 just by looking at that chart and someone throwing random examples at it -.-

disappointing, Numberphile.

I really like his hair.

Convex vs distributed weight gives a majority of global coins the heads up. (if caught in the hand and turned down.)

I'd love to back to school and tell my Maths teacher a coin flip is 51/49. That would rustle his Jimmy's so bad.

I literally frowned when he said feet per second

Why are we arguing so much?

Or the experiment wasn't done enough times and so the probability theory didn't apply?

I like to proof my hand made dices dividing the sides for 100%, give 30% of margin of error and toss 100 times and see if the numbers fit the percent's ranges.

·

A question: Any advise in more acurate percent per sides and/or margin of error?

the throwing speed from human hand is not precise and that s what makes it random. there s also the time u catch it

First of all, that thumb.

Secondly, I can influence coin flips by changing the height of which I catch the coin. That or I'm bat shit crazy

Random = when you don't know the variables to find the result.

Red machine always tosses a head

Blue machine always tosses a tails

Now if you dont know which machine that is tossing, you can call it random.

No Game No Life Anyone?

I can get a coin flip to land the way I want it to with 80%+ accuracy (as long as the choice is made before the throw, and I can choose which side is up before the throw), so the short answer is: not very.

What are the chances of winning if you flipped the coin 1000 times and you changed the side each time? Can i increase my chances of winning if i do this?

Brady, can you please get one of your professors to walk through Bayes theorem, and its implications in applied mathematics?

When I was in elementary school I won a lot of quarters. I trained myself to flip a quarter in the same way every time so that over 90% of the time it would land on the opposite face that it started on. Chocolate milk instead of regular milk for three years in a row just because I taught myself to flip any quarter the same way every time.

I follow your instructions and flipped it to the moon. You owe me a quarter

This is a nice professor 🙂

Nice, I love investigating these simple questions. For you bicycle riders out there with O-locks on your bikes: Have you guys noticed how frequent your lock hit the spokes when trying to lock your bike? The high rate of occurrence caught my attention so much that I calculated the probability for the O-lock to hit the spokes and found P(36) ~ 0.23 for an ordinary bike having tire diameter of .6m and 36 spokes. That's almost 1 in 4 :-O

So if i bet random people on the street 1$ on a coin flip, and let them flip, but i get to make the call.

I could make some serious money…

I read in a text about how you can always get the correct flip if you held your palm as a plane perpendicular to the direction of the angular force on the coin when you flipped it. At least, this is how I recall it being told. There was a whole mathematical proof but if someone has more knowledge on the subject do fill me in. It's been a while since I read this.

Twelve-dimensional space? That sounds like fun. And here I thought that there were only ten or perhaps eleven dimensions.

i do this but i use it to bet we bet 100 dollers on ourselfs and i know on what side to throw it and what strength so i always win i have won over 1k from my friend XDDDD he doesnt care hes a millionaire soz for grammer cant be stuffed correcting

recently I was bored at work and I started to flip an American quarter and catch it in my palm and then flip it over. after only about 3 flips I got tails 8 times in a row. i was pretty surprised.

Why would you have the x axis stand for speed instead of the duration that the coin is in the air? Because if the coin is flipped at different heights the coin would be able to flip more, right?

Well, there's always the off-axis spin trick you can use to rig a coin toss…

no metric system, Thumb downed

Ok, let's suppose that me and my friend decide to toss a coin. If I have a very bad coin the probability of getting head would be 0.9. Let's suppose I have one of them (but I don't know it). I will think that the probability of getting head is 0.5, but in fact, if I toss the coin 10^4 times it is not. My question is: if I toss the coin only one time and I don't know that it's faulty (for example head=0.9), don't I "transfer" the probability of the coin toss to the probability of choosing the most probable outcome? If this is true, practically, every single coin toss would be 100% fair!

Am I wrong? Is there something wrong or is this true?

I'm sorry for my language, I'm not a native English speaker.

thank you hector salamanca

thing is it is not that the tossing of the coin may be random… as one can see in the way that has been done in the film it is not quite random… what is random how ever is the catching process and the landing process: how does the coin end its travel….

A while back I used some old parts out of an R/C plan I built and ended up crashing, to make a coin flipping machine. At one point I was able to get a 70% chance of landing heads. That is, after repeated attempts, and flipping 1 coin 100 times, It averaged 70 heads, and 30 tails. I'm sure someone who is much more skilled in mathematics could do better, but I was pretty happy with that!

Basically nothing is random.

there are some outlying setups that make it very non random.

when I do the coin flip, tossing with the thumb as shown in the video, catching in my hand and placing on the back of my other hand (a common way to display results of a toss where i grew up) with an Australian 20 cent coin it is almost certain (~50 tosses in a row, no counter examples) to be the opposite way up to the starting orientation.

i don't get the same level of consistency with other coins.

I always call heads and 7/10 win i put it down to me just being lucky…

I have seen someone do that exept it was a dice 6 sides not 2 sides.

12 dimensional space O_O What are these dimensions?

Out of 100 starting at heads: average 51, but out of 100 starting at tails: average 51. I was the only in my class who had the reasoning. They thought I was some genius.

when you look at a coin that is being flipped in the air, the side that the light reflects off of is the side that will be up when it lands. same goes with spinning it.

Hi. Maybe someone know how to construct unfair coin, that looks like common coin? Without any external things. Because I can't think out any possible way to create unfair coin.

Could you please use or add the SI measure and imperial ones. We are in the 21st century… Really for the rest of the world it's not funny and these videos I presume they have to be…, thanks

These videos are awesome!

How many coin tosses must one flip before the law of large numbers takes over? Can we say something about a single coin toss? Or only the probability density for a set of coin tosses? How small can such a set be before the prior probability is informative?

This guy always interesting topics but explains them in very disorganized and rambling ways

I wish this video had defined what he meant by "coin flip"

Flip a coin and let it land on the grass?

Flip a coin and catch it in your palm?

Flip a coin, catch it, and slap it on the back of your other hand?

what if you bet 1k dollars if a nickel landed on its head to one if your friends and same for your friend 1k too and on tails and it on its edge? then what?

This guy didn't even look at the camera once x'D

The real question is what's the most you've ever lost on a coin toss.

What about coin tossed from the edge?

How do tou account for catching the coin lower that you tossed it, or that some people flip the coin once more after they catch it?

Next would be what about a magician ? Someone with nimble hands, how predictable can he become ? (Requires some parameters : a player would reject a coin toss that looks too flat or small)

Using the imperial measuring system in a science video is roughly equivalent to using prayers to demonstrate a scientific result…

If you flip the coin once, take whichever side came up and flip it again, its more balanced, because its .51*.51+.49*49 (coming up the side it first started on) vs .51*.49+.51*.49 (coming up on the other side), which is .5002 to .4998. If you do this process again, it comes out to .500004 and .499996. Again: .50000008 to 49999992; .5000000016 to .4999999984; .500000000032 to 0.499999999964. Basically, just do this a few times, and the chance of coming up the same as the original becomes much more even.

The scariest thumb ever videoed.

Y'all are a bunch of wankers. Last time I checked this was a youtube video, not a dissertation. FPS is fine for describing coin flips on youtube.

Are coins usually equally heavy on both sides (sliced so discs would be created) and if not how much could a reasonable weight difference change the outcome of the coin toss?

If you flip it to the moon, it would be 0.49 leaving you with the same side you started with! 😀

People should be more responsible with using the Imperial System.

IMPERIAL AT YOUR OWN RISK.

Wow! Top three comments argue over the use of imperial units in a video that's about distribution. What are the odds? I mean, humans

arequite intelligent, right?Wouldn´t it be more accurate to name te vertical aixis as Torque?

Instead of flipping it, start on the floor and flick it under your finger, then how do you determine the probability of heads / tails when the starting state isn't one of the faces ?

Is that a 1964 Kennedy half dollar you keep flipping

If so that is a pretty special flipping coin you have there 64 was the first year they had Kennedy and it was the last year they were made of 90% silver

This channel is awesome

Nothing is random, it's just that we don't comprehend all the factors, hence we think it's random. Imagine the implications of that.

You stick to our FEET per second, bro. I'm with you all the way! Screw these metric sexuals!

never take possibility of two think virtual phi.

943rd

3:17 I like how his thumbnail is literally bruised from flipping coins so much. Now that's dedication!

Yay! FPS!

OK do dice next. Papa needs a new coat

Nothing is truly random really, we only perceive things like coin tosses, 50/50 chance, as random because it's very very complex for the human mind to predict a coin toss's outcome as it happens.

Depends on the coin.

Only .51? I thought it would be higher.

I find it hard to believe that this bias is the same for all people, I am sure there is a variance. If you hardly give it any rotation the bias will be more, I would guess.

But moreover he didn't include a 3rd typical variable: height!

Do you drop it on the ground, or catch it in the air, and if you catch it, at a higher height, or lower, or does it depend how far away from your body you need to catch it.

Nah if you flip it to the moon the bias is no longer. 51, but almost exactly .50

Don’t be so rude to him guys he is so happy and he’ll feel down if he reads the clmments

This guy is awesome!

How come he didn't take the probability of the coin landing on its side?

0:53

Feet per second : American

Révolutions per second : French

A lot of people, including every NFL referee ever, can't get the coin to turn more than twice in the air.

The question is how could one make a trully random phenomenon without involving electrons?

Everyone else: Why is he using imperial measurements?

Me: Why is he using a half dollar coin?

I prefer a D20 dice over a coin toss. Even numbers heads, odd numbers tails.

They are about to ruin Leonard Susskind's teaching carrier… 🥺😅

I say one flips, the other catches it and the one that flips calls it while it the coin is rotating.