I find probability, randomness and
statistics to be perhaps one of the most inherently ordered and satisfying
areas of mathematics; and despite being a topic that a lot of people have a
vague notion or understanding of, I’ve also discovered that people’s
understanding of it is rife with crucial misconceptions.

Let’s illustrate one of the most
common misconceptions through an example:

Suppose 10
people are going to be asked to insert their hand into a bag and take out a
coloured ball without looking. Inside the bag are 3 blue balls and 7 red balls
for a total of 10 balls. Once each person has chosen his ball, the colour he
selected is logged, the ball is placed back inside the bag and the next person chooses.

Now, after the
10 people have chosen their ball and their colours registered (you were not
present while this was done and therefore don’t know how many times each colour
came out), the gamemaster comes to you and asks: “Do you want to bet that blue
balls were chosen 3 times and red balls 7 times? If this was the case, I’ll
give you 100 dollars but if not you give me 100 dollars.”

Do you take the
bet?

A lot of people would
take the bet, meaning a lot of people would lose 100 dollars. The probability
of you winning that bet is merely around 26%.

One of the most
important theorems in probability, the Law of Large Numbers, states that if you
continue to ask more and more people to select balls, when the number of people
goes towards infinity then the proportion of “blue balls” will converge to 30%,
but it says nothing about finite draws.

Most people wrongly
interpret the probability associated to the example above as: “out of 10 cases,
3 will be blue and 7 will be red”. They fail to grasp that the way in which
probability balances itself out is only in the long run.

Do you think
Casino owners look through the CCTV biting their nails and cursing each time
the house loses a hand? Probably not. They understand the Law of Large Numbers
and that in the long run, the probability layout of their games means they earn
money no matter what happens in a hand or even a whole night.

One of the areas
of football that most excites and interests me is player recruitment and talent
identification. Why is understanding probability important for this? Every big
club in the world wants to quickly identify the superstars of the future at a
young age and bring them into their academy; and to do so they must scan millions
of young players across the whole world and have a system in place that correctly
identifies future talents. However, it doesn’t always work out for all those recruited
players does it? Many simply play their way through the academy system without
their career mounting up to anything of note. How should football clubs
approach their selection to maximise the number of future stars they recruit in
lieu of the unremarkable?

Suppose then
that every player comes in one of two categories: destined to be a future star
and destined for an unremarkable career. The ultimate truth will only be known
after his career unfolds and he retires; but the key conceptual element is to
view him as having been in his respective category all along. This is obviously
overly simplified, and in general you want a model involving more than these
two categories, but for the sake of simplicity in the point I want to get
across, let’s consider this simplified case.

The difference
between which of the categories a player is in can sometimes be as subjective
as “divorced parents”, but this doesn’t make it any less appropriate to be
studied by mathematics. In fact, many

*soft*variables such as these are used in Banks’ models for predicting whether a person is in the “will pay loan” or “won’t pay loan” category.
However,
sometimes these subtle differences aren’t in the available information of each
player. Perhaps the scout trying to decide who to recruit doesn’t know whose
parents are divorced and in the eyes of the information that is available to
him (in-game stats of the players perhaps), two players from the two different
categories can look exactly the same.

The typical
example in mathematics textbooks is classifying a new fish whose species you
don’t know to be either salmon or trout, based solely on some variables such as
its length and width.

Blue points
represent previously recorded instances of trout for these two variables, and
red points represents previous instances of salmon. When a new fish comes
along, you can’t be 100% sure of which category it belongs to, but you can make
an educated prediction by assigning the category that seems more likely from
your previous observations. In the example, the black line divides the space
into two areas: one where you predict “salmon”, and another where you predict “trout”.

Since for the
available information the two categories “overlap”, you must accept a degree of
error in your prediction, unless you include more variables that can better
distinguish the categories such as length of fins or divorced parents.

This “inevitable
error” is called the

**Bayesian error**, and the line dividing your classification into categories the**Bayesian decision boundary.**Making decisions using the Bayesian boundary is the best*decision making model*for these type of problems, and the Bayes error is the lowest level of error any model can aspire to.
Bayesian
decision boundaries can be used to make informed decisions on player
recruitment. If we involve as much information as we can collect on players
such as in-game stats and even softer variables about his personal life, we can
use the info on previous “

*stars*” or “*unremarkables*” to produce a Bayes decision model and predict new players into one of these two categories. This sort of approach has the advantage that it can process millions of players quickly, economically and efficiently; without the need of a huge scouting network or machinery. Most clubs would rebuff the idea of liaising their scouting process 100% to these methodologies, but at least it can be used first as a filter to narrow down millions of players into a selected handful that the human component of the scouting process can analyse.
The problem these approaches face is that when things are new to town
they have a point to prove, the opposite of “innocent until proven otherwise”. Disregarding
the Bayes error associated with scouting, these methods are expected to be magic
crystal balls and are only given a couple of chances before their fate is
sealed by these outcomes. It’s as if a Casino Group come to you proposing you
invest in a new Casino and you say to them: "OK,
get me one of your croupiers and let´s play a hand of blackjack, I want to see
if it´s profitable”. If the house loses the hand, do you decide against the
investment?

On the other hand, I read on the news the other day that Arsenal signed
the Leicester scout that recruited Jamie Vardy, as if this single outcome is a
sure sign of sustainable success.

I can’t assure you that you won’t miss out on a great player by using this
approach, or that every player you sign will be fantastic; but the Law of Large
Numbers ensures that these approaches will represent sustainable success in the
long-run.

The true problem is convincing such a short-term industry to embrace it.

## No comments:

## Post a Comment