England's World Cup Record Isn't Bad—Math Proves It

[00:00] So every major football tournament, England football fans are told one thing, "It's coming home. It's coming. Football's coming home." But by the

[00:12] end of the tournament, uh, realistic expectations have set in, and England And because of the number of tournaments that we have entered, everybody

[00:24] So my question to you, that I'm gonna answer in this video is, of hurt, but 60 years' worth of typical mathematical expectations?

[00:39] And that's what I'm gonna explain in this video I mean, after all, English football is the richest in the world.

[00:56] Um, we regularly produce sort of what look like decent teams, but somehow after 60 years, we've only managed to win one World Cup. should we have done a little bit better?

[01:11] But where does that reality sit in terms of true expectations? Because every time you enter a tournament, there will be a set of odds. If we look at the current tournament, you can see what the set of odds

[01:25] represent an implied probability. So on this particular occasion, at this particular tournament, England were set up with about a 10% chance of winning the entire thing.

[01:40] So given that England have got a 10% chance of winning, you know, Is there a sort of a pattern that appears? Is there … Yeah, because you can't sort of go into a tournament expecting to win.

[01:53] You'd have to say, well, Argentina have got this, uh, chance of winning, Brazil have got the chance of winning, Spain, and so on. of England winning a tournament.

[02:09] Uh, we can go back over a number of years, average all of that out, and you end up with this particular table, which is the expectation that on any given, uh, World Cup on average, this is the chance that those international teams

[02:22] So when we have that data, we can begin to work out things like how many World Cups could England have won in the last 60 years? And for that, we need to go all the way back to 1654.

[02:38] So why have England only won one World Cup, and really should we have won more? Well, in order to answer that question, I do need to get in my time machine, So let's do that.

[02:58] Okay, so if we go back to 1654, uh, we could have gone to France and we would have seen two mathematicians, Blaise Pascal and Pierre de Fermat, They were looking and trying to understand could you actually predict

[03:12] There seemed to be no structure to them. Was there a way of mathematically describing exactly how this would occur? And essentially, uh, Blaise Pascal and Pierre de Fermat, uh, were the

[03:27] we use every day now, especially when we're looking at sports and betting Um, so they kickstarted that, but it was a few years later that we had Jacob

[03:40] Bernoulli who actually narrowed that field down to a specific problem, and he was looking at where there are two outcomes, a true or a false or, or a head and a tails.

[03:52] Is there something that I can describe that beautifully describes exactly, you know, what occurs when we have a yes or a no, or a head or a tails?

[04:04] And can we describe, you know, what is the chance of that over discrete periods, um, and how can, how can we actually look at that, uh, And he came up with the concept of binomial distribution.

[04:18] I'm sure you can go off and research that yourself. to use to describe the chances of England winning the World Cup, and indeed, how many World Cups should they have won over a period of time.

[04:34] Because, uh, Bernoulli also, uh, came up with a theory about how if you repeat an experiment enough times, it does tend to converge onto its true But yeah, wanted to pay simple homage to the mathematicians that created

[04:50] what we're gonna use to work out how many World Cups should England have won since the World Cup started. If you have a, an event occurs and an event does not occur, then you can use a

[05:06] And when we're looking at football and the World Cup, of course, we do have that. We ha- we're saying, England win the World Cup, England do not win the World Cup. So we've already worked out one of those key variables, which is what

[05:22] is the chance of England winning any sp- uh, specific World Cup? On average, whenever the tournament is created, England have about a 10% chance

[05:34] Uh, but then we also need to know how many iterations of that do we go through? Do we put them in 5, 10, 15, 20, however many, uh, they have qualified, and so on.

[05:47] Uh, but if we combine those two things and what the binomial distribution will do, it So if we simplify this in terms of a coin toss, if we toss a coin, The sort of question we're trying to answer is, you know, what's the

[06:01] chance of us just getting one head in five coin tosses or one head in 20? Um, or, you know, could there be two or three or… And that's what the binomial So taking those stats, we can put them into a nice graph and actually

[06:17] explain what is the true chance and how many World Cups should England have won in all of the tournaments that they have actually participated in. can look at the results of it.

[06:30] On the Y-axis, you can see the number of tournaments that have taken place. we're looking at has competed in. Brazil have competed in 22 World Cups.

[06:46] Across the X-axis, you actually have the number of World Cups that you should have won, given that you have competed in this number of tournaments. actual percentage chance of winning any one particular tournament.

[07:03] Brazil have competed at 22 World Cups, and when we looked at their odds from recent tournaments, it turned out that they averaged about a 20% chance

[07:16] So if we put tw- uh, 20% in, we look at that line with 22 on it, and then we move across, we'll see that the chance of them winning

[07:28] four World Cups is where that peaks. they have won five World Cups. here, is only a couple of percent.

[07:41] So really, what this chart is telling you is that Brazil should have won four or five World Cups, and they have actually won five. So yeah, it's a pretty good approximation.

[07:53] So all that I've changed, um, on this heat map now is that percentage chance We've moved it from 20% down to 10%, and this more or less is sort of reflective of what you would think would apply to England.

[08:08] Every time they compete in a World Cup, they've got about a 10% chance of winning it, which is exactly where we've ended up this year. But the interesting thing is that while Brazil have competed, uh, 22

[08:20] So if we actually look at that line of 16 and then go across, you can see the highest value is actually under one World Cup. It's basically saying that given 16 appearances and a 10% chance of

[08:34] winning the World Cup in any one particular tournament, England should have won one World Cup, which is exactly where they have ended up. The chance of winning zero is, uh, quite significantly smaller, and the

[08:46] chance of winning two is a little bit closer to that, but it's still some So yeah, uh, Jakob Bernoulli successfully correctly predicted that England would have had 60 years of hurt.

[09:01] So if you go back and look at the graph I produced earlier, based upon most recent World Cups, England have had an 8.69% chance of winning those. and that's more reflective on the construction of teams in the modern game.

[09:17] Um, but you can see if we use 8.69 as the base, uh, for this particular heat map, and we look at 16, again, it really sticks out that one is the answer. We should have won one Gold Cup from all of the 16 appearances,

[09:32] Um, so yeah, you know, we can either increase the chance of winning each time or play a lot more World Cups if we want to win that coveted prize.

[09:44] So when we use the binomial distribution, it gives a good description of And when we transplant that onto the World Cup, we have to look at two things: the chance of any individual team winning any specific World Cup, uh, we've averaged

[09:59] those odds out, and then how many times have they competed in the World Cup. And therefore, for teams like Brazil, then they've competed in 22 World Cups, and their chance of winning on average is about 20%, and that comes out really close

[10:14] They've won five World Cups, and the binomial distribution is saying, you And of course, with England, when we plug in that exact percentage, that

[10:27] 8.69 that we've taken from recent tournaments, then that actually strongly suggests that we should have only won one based upon 16 experiences. So yeah, while it is 60 years of hurt, um, it is actually mathematically

[10:41] Um, so the answer to England winning more than one World Cup is to have a much So I'm pretty sure there will be a second one at some point,

[10:54] And unfortunately, that is one question that I'm unable to However, I do hope that this video was useful for you.