[0:08] [music]
[0:08] Eight years ago, a younger version 
of me sat down and flipped a coin  
[0:13] 10,000 times. If you're wondering 
why would anyone do that,  
[0:18] look, we don't welcome those sorts of questions 
around here. But more to the point, I want to see  
[0:23] how often it would land on its edge. And later on, 
I'll explain why I was doing that, how many times  
[0:31] it landed on its edge if you want to see if you 
can have a guess. 10,000 flips, how many edges,  
[0:35] and I have a ridiculous joke collaboration we 
shot, which has never seen the light of day.  
[0:41] That'll be right at the end of the video. See if 
you can guess who else is involved. But for now,  
[0:46] we're going to speed forward 8 years to 
when these flips finally became useful.  
[0:52] That's because of Pi Day. Yes, a few years ago I 
flipped a coin 10,000 times. I don't know what I  
[0:57] was thinking, but I've finally come up with a use 
for that footage. I'm going to use it this Pi Day.  
[1:02] So, hey, happy Pi Day 2026. Although, shouldn't 
have said that. Forget [music] it's Pi Day.  
[1:07] Spoiler. This is not a video about that. 
This is a video about flipping a coin to  
[1:13] about flipping Thank you. a coin to see 
what the ratio between heads and tails is.
[1:28] Yes. Late last year, a new bit of math 
dropped relating to if you flip a coin,  
[1:33] keep track of if it's heads or tails, and you 
stop when there are more heads than tails.  
[1:39] And let's say you do that over and over. And 
then you ask the question, hang on a second. If  
[1:43] I stop whenever there are more heads than tails, 
what's the average ratio of all those sequences  
[1:48] where I compare the number of heads to the 
number of tails? And there's going to be  
[1:51] more heads than tails, but how many more? And if 
you calculate that and work it out in general,  
[1:59] turns out, and yes, this is why it's a Pi 
Day video. And look, I'm as angry as you are.  
[2:04] It's pi. It's pi on four. If you flip a coin until 
there are more heads than tails, the ratio is pi  
[2:14] on four. So that's what we're going to do. This 
Pi Day, I'm going to calculate pi from my 10,000  
[2:19] coin flips, which has actually come together quite 
well. Previously, this Pi Day was going to be  
[2:24] us calculating pi on the moon. You may have seen 
the update about that that went out recently.  
[2:29] Sadly, like a lot of space missions, it's been 
delayed. So, we can't do that. That would be next  
[2:33] Pi Day hopefully. Previously, previously, 
there should have been a Pi by hand year.  
[2:39] I was going to try and break the record. Turns out 
that's going to take longer than I expected. So,  
[2:42] that's also delayed, but will happen. link in the 
description if you want to sign up for updates.  
[2:47] So, what I'm going to do is basically go back to 
normal. I'm going to do a standard pi by hand,  
[2:51] but this time instead of doing long tedious 
working out by hand, I've previously flipped  
[2:56] a coin 10,000 times by hand. And from that, we 
can extract a value of pi. And unbelievably,  
[3:05] no one noticed that you get this value of pi on 
four from multiple coin flips until last year.  
[3:12] And so I'm like, well, this is great. Breaking 
news. So here's what we're going to do. I'm going  
[3:16] to explain why you get pi when you flip a coin. 
Then we're going to get pi from flipping a coin.  
[3:23] Let's do it. Right. I'm going to use these to 
represent a sequence of heads and tails. We  
[3:27] can kind of arrange them across here. 
And actually before we even start,  
[3:31] we've got a pretty tight range on what the average 
ratio can be. Because the highest it can be is if  
[3:37] boom, straight out of the gate, you flip a heads. 
You've now got more tails than heads as required  
[3:42] and it's 100% 100% heads. It's the highest 
possible ratio. Or we could have like, you know,  
[3:48] thousands of flips before eventually we have 
more heads than tails. And when that happens,  
[3:54] it will have just because there's so many just 
gone over 50%. So really our range is 0.5 to one.  
[4:02] Because we're calculating pi on four, 
it means we've already narrowed pi down  
[4:07] to between two and four, which you know, compared 
to some historic pi days, we're doing pretty good.  
[4:16] Now, what if we didn't get head straight 
away? What if the very first flip was tails?  
[4:20] Well, now we're going to have to keep going. 
And the shortest sequence it could be now is  
[4:24] three flips. It can't be two because 
the best case scenario is we get heads,  
[4:30] right? And but now it's 50/50. Has to be more 
has to be greater than 50%. So potentially  
[4:37] another heads. So now that's the Whoops. Shortest 
sequence where it goes tails, heads, heads, stop.  
[4:46] And that's a well, I guess that's a a half time a 
half times a half. That's a half cubed probability  
[4:52] of happening. And the ratio is two out of three. 
But what if we didn't get that second heads?  
[4:59] The next shortest would be five flips. So 
for us to keep going in the first place,  
[5:04] that would have to be tails there. Okay. And then 
the next one will Oh, it has to be two more heads,  
[5:12] right? Cuz we need to get it needs to be three 
out of five heads for us to stop after five.  
[5:17] So it would look like that. There it is. Okay. 
Next shorter sequence is five. It ends on when  
[5:22] we flip this heads. And that's the one that 
tips us over to more heads than tails. So the  
[5:26] probability of this sequence happening half times 
a half times over five of them half the power of  
[5:31] five. We can add that in and the ratio is 3 out of 
five. Great. So we've Oh, hang on. Wait a minute.  
[5:38] Yeah, this one works. But so does so is this 
one. That's also a perfectly valid sequence.  
[5:45] That's five long and would end at the same 
point exactly when we flip that one. So there's  
[5:51] I'll pick that up later. There's I can replace 
it. So there's there's two of them. Okay, so now  
[5:57] for each run of coins, we need the probability, we 
need the ratio, and we need how many options there  
[6:04] are. So we multiply this one by two. Let's have 
a go at seven. Right, I've set us up seven spots,  
[6:12] and we're going to think about what sequences 
could possibly end at the seventh flip. And  
[6:16] by end, it means that has to be us flipping a 
head because that's the point at which we have  
[6:21] more heads than tails and we stop. And at no point 
before then can we have had more heads than tails  
[6:26] otherwise we would have stopped including the very 
first flip. There has to be a tails otherwise we  
[6:30] would have stopped on the heads. The second one 
could be heads or tails. We'll come back to that.  
[6:35] Let's start by the case where we flip three tails 
first. So tails are way out in front and then  
[6:41] heads come slamming back and they get it right at 
the end because four out of seven is the point at  
[6:49] which this one means we've got more heads and 
tails. So that's one. Write that down. However,  
[6:55] this third tails here, that could happen later. 
We could have that one was a heads and that one  
[7:00] was a tails. That would work nicely. We can move 
it further. That's two. If you're counting, we can  
[7:05] move it down again. That could be uh heads. That 
could be the tails. We can't do it again, however,  
[7:12] because if this was heads and that was tails, 
it doesn't work because we've already now look  
[7:18] at that. Two tails, three heads, we would have 
stopped. So, we only get those two extra ones.  
[7:23] So, now we're up to uh three alto together. Now, 
the other case was if that was a heads there.  
[7:30] Now, if that's a heads, this one can't 
be heads cuz otherwise we'll have more  
[7:34] heads than tails. It's got to be tails like 
that. And then that's still heads over there.  
[7:39] And then these can go either way. This is either 
a heads than a tails or it's a tails than a heads.  
[7:45] Both of those work. You can't delay it any 
longer. So, that that's five. If you want to  
[7:50] pause and go through that yourself, you're very 
welcome to. I spent a long time in my notebook,  
[7:55] which I'm pointing down here because I've still 
got all my This was me trying to work it out  
[7:58] the other night, and I got all these sketches 
of me going through being like, "Oh, what do I  
[8:04] get in the sequence of heads and tails and tails 
and heads? There's my there's me counting five,  
[8:09] and I was right." So, I was able to fill that into 
the table. And the next one, we could do it for  
[8:15] nine. There's 14 ways to do it. I didn't bother 
working that out. I put the sequence of 125 into  
[8:26] the online encyclopedia of integer sequences and 
it was all like, do you mean the Catalan numbers?  
[8:33] And I was like, turns out I do. So why the 
Catalan numbers? Well, to get my head around it,  
[8:40] I then drew a diagram which I'm going to recreate 
for you very quickly here on my whiteboard  
[8:45] where I thought, you know what? Let's represent on 
a kind of tree diagram. If you get a tails, you go  
[8:51] down. If you get a heads, you go up. I had to roll 
that in around the dog so I didn't hit the dog.  
[8:57] And the goal is to end up above water. Very 
simple. So, you start here on the surface  
[9:03] and then if your first flip is a heads, you're 
high and dry, you stop. If it's a tails,  
[9:10] you're down here. To clarify, when 
I mean above water, we stop flipping  
[9:15] once we're one above this blue line. Because 
if tails moves us down and heads moves us up,  
[9:21] being one above our starting point means we've 
now had more heads than tails. And then I just  
[9:27] expanded the tree diagram because the next 
one would bring you back up to the surface.  
[9:32] This one would bring you even further below the 
surface. Then you've got a chance to get out here  
[9:37] heads. And there's only one way that that's 
one way to do that. There's one way to do that,  
[9:42] but there are multiple ways. If you go up here 
next or down there and then up there and then out,  
[9:51] then if you count that, you're like, "Oh, actually 
there's two ways because I could have gone all the  
[9:53] way down and all the way back and out or down, up, 
down, up and out." And so if you start building up  
[9:59] this diagram of all the ways you can flip 
heads and tails and all the places you can end  
[10:06] up and all the times that you break the 
surface and escape and then you got to  
[10:11] count the number of paths. Ah, you know what this 
is reminding me of. But if we flipped it over,  
[10:19] it's Pascal's freaking triangle. If you'd like 
to learn more about Pascal's triangle and the  
[10:23] Catalan numbers, my friend Sophie Mlan already 
has a fantastic number file video all about it. I  
[10:28] highly recommend you check that out. And it works 
very neatly because Pascal triangle is effectively  
[10:33] counting the number of ways you can get to any 
given point. And we can use the Catalan numbers  
[10:37] to normally cross over in the middle. Ah, it's 
all it's really nice. Check out Sophie's video.  
[10:43] But for our purposes, we're just going to 
use the fact that the Catalan numbers counts  
[10:46] the number of possible head or tail sequences. 
Right now we're going to put it all together.  
[10:52] So we can take our table from before and put it 
into a generalized form. Catalan numbers start at  
[10:58] n equals 0. The zeroth Catalan number which is 
1. So we're going to use counting from zero.  
[11:05] And the formula in general is going 
to be 1 / 2 to the power of 2 n + 1  
[11:12] multiplied by the nth Catalan number multiplied 
by n +1 over 2 n + 1. That's just what we had in  
[11:21] the table. But now you put in whatever value of 
n and you get the next term in the series. And so  
[11:26] we want to sum these from n equals z up infinitely 
many of them. Then we get some result. And we do.  
[11:35] I wrote down here last night. You can see there 
was my terrible attempt at that tree diagram. Um,  
[11:39] and over here next to it, I've written the formula 
for the catland numbers. 1 / n +1 multiplied  
[11:46] by 2 n choose n. And you can put that in terms 
of factorials. We just substitute that in.  
[11:53] And now we have one big chunky equation that 
gives us each of the terms from n equals z  
[12:01] all the way up. We just got to work out how we 
turn that into pi. Okay, now we just need some  
[12:09] kind of substitution or a way to rearrange 
this to get closer to pi. What's that noise?  
[12:17] It's arc sign. It's the inverse 
function of sign that can be written  
[12:23] as a series. Look at that. Isn't that amazing? I 
mean, okay, look, here's the thing. I can't derive  
[12:29] everything from first principles. It'll be a very 
long and I'm aware of the things I'm about to say  
[12:34] boring video. So at some point you've just got to 
take something on faith and we're going to use the  
[12:40] series expansion of aride. If you want to look 
up how it's done, you can. In fact, it involves  
[12:45] binomials and once again Pascal's triangle. Lovely 
bit of mathematics. That's now your problem. All  
[12:52] I'm saying is doesn't that look similar? In fact, 
instead of ark sine of x, we could do ar sign of  
[12:59] one. Just put that in. Why not? And that means the 
x becomes one. That whole thing vanishes. And it  
[13:05] looks pretty much exactly like what we had before. 
In fact, it's just twice as big. It's two times  
[13:14] whatever the average ratio of heads to tails is 
in our previous calculation. And what's the arc  
[13:20] sign of one? The inverse sign of one. What value 
gives you one when you put it in sign? It's 90°.  
[13:27] It's pi on 2. So pi on 2 equals twice the average. 
The average is pi on four. It's still making me  
[13:35] angry. I mean, here's the thing. We're not 
surprised that if you flip a coin lots, the  
[13:42] dogs come over to see why I'm getting emotional. 
Uh, we're not surprised that stopping lots of  
[13:48] different length sequences of flips and adding 
them together gives us a series. That's what  
[13:52] we'd anticipate and we're not surprised that you 
can represent pi with an infinite series. That's  
[13:58] classic pi. But I expect them to be separate 
things. What stuns me is that one of the series  
[14:06] for pi happens to match a series for flipping a 
coin. It's ridiculous. I'm not happy about it.  
[14:16] Thank goodness I got my therapy dog right here. 
Oh, who's a good dog? This ridiculous pi coin fact  
[14:22] was discovered last year by a mathematician I know 
named James prop. And James is going to put up a  
[14:27] blog post for this pi day going through all the 
mathematics behind it. So I'll link to that below.  
[14:32] You can check it out if you want to go through 
the detail. It's even a bit of calculus in there  
[14:37] if you're into that kind of thing. But 
for me normally my problem would be oh I'm  
[14:42] like a such a good fact. I need to do a thing 
for pi day. I could sit here and flip a coin  
[14:46] over and over and over again and calculate pi by 
hand. But this time, I don't have to because it  
[14:52] turns out it's already been done by my longtime 
collaborator, past Matt. So, here's why past Matt  
[14:59] was flipping a coin 10,000 times. I've had this 
longunning project in my creator calculate how  
[15:06] thick a three-sided coin would have to be, which 
is a coin where it's equally likely to land heads,  
[15:13] tails, or edge. And I've talked about this 
before. or it's been on hold for a long time,  
[15:18] but I found the current state-of-the-art. 
But I realized the state-of-the-art was a  
[15:23] single paper. In fact, to this day, if you go to 
Wikipedia and look at the entry on coin flipping,  
[15:28] the one citation for landing on edge is the 
paper probability of a tossed coin landing on  
[15:35] edge from 1993. So, I thought I would recreate 
their experiments and I'll do my own experiment  
[15:44] in an attempt to investigate that. 
Now, I never got around to analyzing  
[15:48] the data. I did recreate their experiment by 
sliding the pound coin, which is what they used,  
[15:54] off a horizontal surface a thousand times. Then, 
I did my own 10,000 flips. I'll link to it below  
[16:01] if you want to have a look at the data. You can 
analyze it yourself. Uh the big reveal 14 times  
[16:07] landed on its edge 14 times. Now this whole 
experiment was a complete waste. It became  
[16:13] a whole section in my book humble pie. But the 
footage the footage I never had a reason to use  
[16:20] until now. Until I wanted to calculate pi. 
So here's my big spreadsheet of data. I'll  
[16:26] link to this below if you want to check it out 
for yourself. It's got the exact time I flipped  
[16:29] every single coin. Wow, that was a late night. 
And then over here, started heads, landed heads,  
[16:35] every single landing one. And that's just my kind 
of count count if 14 edges. Ah, good times. Now,  
[16:43] we need to analyze this, which means we got 
to get into Python. And I thought it might be  
[16:47] fun to write the code. So, all I'm doing here is 
I'm using open py. That's Python for Excel files.  
[16:55] I've I've I've set up the name of the file and 
everything. We want to count how many sequences  
[17:02] we've been through and that starts with 
zero of them. I try to have remotely  
[17:06] useful names. That's also zero. And then for each 
one, we need to count two things. How many heads  
[17:14] and how many flips? What are 
we going to do with edges?  
[17:20] Let's ignore them. [laughter] Let's say they 
don't they don't count. They don't count. It's  
[17:26] got to be heads or tails. Lowerase H or T. So what 
we'll do now is if the flip result equals heads  
[17:38] then we increment head count. So that goes up by 
one. And then if flip result doesn't equal e then  
[17:55] that's our total number of flips. Okay. So, all 
I've done there is each time we flip it, if it's  
[18:00] heads, we add to the running head count. If it's 
not edge, we add to the running total number of  
[18:06] flips count. Great. Now, we got to check if we've 
hit the end of a sequence. So, if that happens,  
[18:13] we want to work out the ratio of heads to tails in 
that run. So, we're going to say this ratio equals  
[18:19] uh headcount divided by flip count. I could 
do all this in one big long chunky line.  
[18:25] I like to split things out because it makes my 
life a little easier to try and read it. So,  
[18:29] I'm going to take running ratio total add on 
this ratio. And we've had one more sequence.  
[18:35] So, uh what do we call our sequence count? Count 
sequences. Well done. Pass mat always there for  
[18:42] me. So, we add one onto that. Now, we want to 
reset cuz we're going to start another run. So,  
[18:48] flip count and head count now get set back to 
zero. Okay, that should be everything. Oh, that's  
[18:55] I can't increment flip result. That's 
what I It's flip count. What a dingus.  
[18:59] Okay, that fixed. I kind of want to check it's 
working before we get the big result. Let's run  
[19:07] it first without calculating the result. That's 
the big reveal. Each time we'll print the result  
[19:14] and then down here we'll print what we think 
the ratio was. Okay, so now we can run it.  
[19:21] It'll do that for all 10,000 flips. make sure it 
functions and make sure it's doing what we expect.  
[19:26] You never know with code. It's like if you just 
run it without making sure you've gone through to  
[19:32] make sure everything's working properly. It's 
a very opaque process which makes me nervous.  
[19:37] Ah. Oh, there it is. It ran. Oh, these 
are long runs. Oh, okay. It is working.  
[19:42] So, that was a super long run. Look at it. And 
when it finally ended, it was just over a half.  
[19:48] Next one was a heads. So, it was one heads was 
one. So this was a total of 1 2 3 4 5 6 7 8 9  
[19:57] 10 11 flips of which one two three four five six. 
So yeah, so it was five and five. One more heads  
[20:09] tipped it over to six and five. It stopped and 
ratio 54.54%. Okay, I'm happy the code is working.  
[20:17] I'm going to undo that one. Now once 
it finishes running, let's just print.  
[20:27] Should we do it all at once? Should we just print 
pi? Let's just print pi. You know what? Let's do  
[20:35] pi val equals running ratio total divided by count 
sequences. So that's the total of all the ratios  
[20:44] divided by how many there are. That gives us 
the average. And then we multiply that by four.  
[20:49] And then we print pi val. So this is it folks. 
This is the value of pi for pi day 2026.  
[21:00] We're going to zoom. Let's get right in. I know 
that's messy, but I want a big number. Okay.  
[21:04] Ready? Okay. Here we can't see producer Nicole is 
just out of shot over there. Very excited. Camera  
[21:13] person Alex is over there wildly indifferent. 
And now we run the code. 3.2266. [music]
[21:24] That's better than I was braced for 
cuz it converges so slow. Wow. 3.2266.
[21:35] For everyone who keeps a running record of 
my successes and failures, write that down.  
[21:42] That's good stuff. Obviously, while I was 
flipping a coin 10,000 times, I had a bit of fun.  
[21:47] did some ridiculous bits to camera and then I 
thought, you know what? Why don't I try this  
[21:51] in a fake collaboration where a coin 
unexpectedly keeps landing miraculously  
[21:57] on its edge. In a previous video, I said there 
was a trivial chance when you flip a coin  
[22:02] of it landing on its edge. And even though you 
can balance a coin on its edge, it's a common  
[22:07] misconception that that can happen when it's 
in motion. If you actually flip a real coin,  
[22:13] it lands on a physical surface. It will 
never land on its edge. And to prove why  
[22:18] that's impossible, you need some very nice 
mathematics, some interesting physics,  
[22:23] and some things you might not know. The first 
step, however, is just to actually flip a coin and  
[22:29] take a close look at what happens. If you watch 
what actually happens when you what? Oh, come on.
[22:41] Okay. Okay. All right. Okay. About about the 
collaboration. I got it. It landed on its edge.  
[22:47] No, you didn't. No. Seriously. Seriously. I 
flipped it. That's not possible. First take  
[22:51] I landed on its edge. Is we pro? We No, I know. I 
know. I know what we proved, but I'm just saying  
[22:56] it happened. Do it again. There's no Look, 
there's no way. What? You You show me then.
[23:07] See? No. Yes. Wait. No. Yeah. I don't 
care what your physics says. It can't  
[23:12] happen. I know. I know. But I did 
it again and it's bang on the edge.  
[23:17] Okay. You're kind of the worst 
time. It's ended on the edge.  
[23:23] I've got some terrible news about our collab. 
[laughter] It's been literally minutes.
[23:32] See you later.
[23:38] Yes, Steve Mold and Tom Scott did stand around 
waiting for as long as it took for me to get that  
[23:45] coin to land on an edge. Absolute legends. Sorry 
it took me 8 years to get it out, folks. Uh but  
[23:51] here we are. And now back to Modern Matt for the 
final wrap-up. And we're done. Ah, it's so great.  
[23:59] 2026 literally 26 in the digits of pi. So write 
that down for the next 12 months. Pi equ= 3.2266.  
[24:08] Ah, fantastic. Now, some updates for the people 
who watch right till the end of the video. Yes,  
[24:14] we will do pi by hand again. We've done some 
testing. We got a new method. We've done a bunch  
[24:20] of research. We got a whole team on it. It's going 
to take it's going to take so many people for so  
[24:25] long. I'm going to need so many volunteers that 
are all prepared to sit there for a while. And we  
[24:31] haven't really got a plan for that. We're trying 
to work out the best way to do it. If Moonpie is  
[24:36] a success on the first launch, the leftover 
funding from that will go into Pi by hand,  
[24:41] which means we can probably do it at the scale 
it requires. I promise I will give you at least  
[24:47] 12 [music] months warning if you want to come 
along and be part of the pie by hand calculation.  
[24:52] Link in the description for the sign up form. 
I will keep you in the loop and I will tell you  
[24:56] more than a year in advance when we're going to 
attempt it. So don't panic. You will be there.  
[25:01] Moonpie, the update is out. That's a lot of fun. 
Go check it out. Link to it below. Hopefully we  
[25:08] will launch um at some point this year. Um we'll 
see what happens. Uh and [music] just in general,  
[25:15] thanks for watching the videos. Oh my goodness. 
This is a wonderful tradition calculating pie.  
[25:19] And I couldn't have done it without Passmat. 
Big thanks to Passmat. Um thanks to Sophie Mlan  
[25:25] for all things Catalan numbers. Jim Prop for 
discovering this ridiculous thing. I'll check out  
[25:31] uh I'll link his blog post. You should check that 
out. Uh, and of course, Steve Mold Tom Scott for  
[25:36] for waiting for a ridiculous amount of time for 
a dumb joke that I didn't release for 8 years.  
[25:43] That's a pretty accurate representation of what 
it's like um to be friends with me. Um, so that's  
[25:49] it. Thanks. Thanks for supporting the videos. 
Thanks for everyone on Patreon. Uh, you know,  
[25:54] I feel like Pi Day is like uh end of one year. 
You know, you get like financial years and school  
[25:59] years. I run on pi years. So, that was the end of 
the 2025 2026 Pi Day season and now we're going to  
[26:07] go uh into the next one. And so, um that's it. So, 
I guess we don't need this coin until next time.
[26:22] Oh, no. [laughter] Oh. Oh, that's so unfair.