Football is the most important unimportant thing in the world — or so they say.
The game, with roots in England, suddenly became globally popular and carved itself into the hearts of people across the world. Over time, football became the most popular sport, a way of life; clubs were founded, stadiums built, fan groups established… Football simply became much more than a game.
Now comes the key question: what does football have to do with the eMatematika, the first Croatian platform for online tutoring specialized in ‘instrukcije iz matematike‘? In its beginnings, football consisted of coaches and players; over time, doctors, physiotherapists, and others joined. However, once football became globally popular, people increasingly began to engage in tactics and analysis. They approached it statistically, and eventually this developed to the point that today every serious football club, in addition to coaches, has numerous analysts and statisticians who advise the coach and provide insights into the potential of certain players in particular formations.
Researchers from Oxford carried out several highly interesting studies analyzing the chances of scoring from different game situations. As most know, in football a penalty kick is considered the harshest punishment and in the majority of cases results in a goal. But mathematicians ask — how often exactly? At Oxford, they created a mathematical model, which was later “tested” in practice. Penalties result in goals 72–77% of the time, depending on the quality of both the shooter and goalkeeper (their ratio matters, and in league matches this tends to even out over time). Free kicks close to goal are also considered dangerous. For free kicks taken within 25 meters, the chance of scoring is about 3.8–4.2%. The probability is higher if the shot goes directly on target. Based on the analysis of 15,000 corner kicks, a model was built projecting that the chance of scoring from a corner is about 3–3.3%. In other words, one goal is expected for every 30 corners taken.
Let’s take penalty kicks as an example. How might we set up a model in this case? Most of us would likely think of the following: the goalkeeper can choose to dive left, right, or stay in the center. Likewise, the shooter can choose to aim left, right, or down the middle. And yes, this is a good and solid approach, since with a small number of variables it covers many possible outcomes. Let’s define X(a,b) as the probability of scoring, where “a” is the side chosen by the goalkeeper, and “b” the side chosen by the shooter. By convention, sides are taken from the goalkeeper’s perspective — left or right. Then we can expect relations such as X(L,R)>X(L,L)<X(R,L) and X(R,L)>X(R,R)<X(L,R). Analogously, this can be set up when either stays in the middle. Furthermore, we want the scoring probability to be equal regardless of which side each player picks. Why? Because otherwise the shooter would always aim at the statistically better side, but over time goalkeepers would figure this out and start saving penalties. The shooter would then change strategy, but his chances would still average out (too much success on one side leads to too much failure once the strategy is exposed — hence, shots to the left and right must be evenly distributed). Imagine a teacher who always calls on students in order by the class roll — those at the end would never study. But what if the teacher suddenly changed strategy?
By setting up equations from the above conditions, we reach the theoretical framework. The next step is to determine in practice the values of the constants involved. First, we need to calculate how large a sample of observed penalties is sufficient to present reliable results. Here concepts from statistics come into play, the most important being variance (loosely, how spread out a dataset is from its average). The Oxford researchers took a sample of 1,500 penalties over 5 years. Some key findings were that it doesn’t matter whether the shooter is left- or right-footed, and that penalty conversion depends slightly on the current score, player pressure, and the presence of spectators. These influences are called external factors, and they may shift conversion rates by only a few percentage points. Ultimately, the conclusion is that penalties result in goals 74.5% of the time. Accounting for statistical error/deviation, this yields a range of 72–77%.
If you are interested in the detailed process of calculating probabilities in sports, visit eMatematika.hr, where we can guide you through many fascinating materials. Who knows — you might be the one to become your favorite club’s future analyst, with the help of personalized math tutoring.
eMatematika Team
