This is a modified version of a post from March 2011 and is dedicated to David Moyes, a misunderstood genius just like me ;)
In interviewing scores of people for jobs in schools over the years, I
have met many people who were quite happy to admit they felt uncomfortable with
numbers, and mathematics in general. My aim in this particular post is to
explain some probability calculations in an accessible and readable way, and
try to demystify some of the supposed difficulty.
Here’s
a question: "What are the chances of Man Utd playing Chelsea AND Real
Madrid playing Barcelona in the Champions League Quarter-Final draw?"
(Doesn't matter which team is at home in the first leg.)
The classic quarter-final draw has eight balls in one pot, which can be
drawn out with no special seedings or restrictions, to give four ties, with
home & away teams decided by the order of the draw. The mathematics
is exactly the same for the current Champions League quarter final draw and the
Sixth Round of the FA Cup or the FA Vase.
Whatever happens, the chances of it happening will be 0.059523809%, and
hopefully this post will prove it in a way that is relatively easy to understand.
Step
One – How many alternative draws are there?
Imagine the balls are numbered 1 to 8. They could emerge in the
order 4>3>7>1>8>5>2>6 but this is only one of many
possibilities.
To establish the pattern, let’s simplify things and consider a draw of
only two teams. There are TWO possibilities for the first ball out, 1 or
2. However, whichever one comes out, there is only ONE ball left in the bag.
So the only possible sequences are 1>2 or 2>1 and a total of 2 x 1 = 2
possibilities.
For three balls in a bag, there are THREE different possibilities for
the first ball out. For EACH of these outcomes, there are TWO different
possibilities for the second one out, and then the third is fixed because there
is only ONE left. This means 3 x 2 x 1 = 6 possibilities. They are
123, 132, 213, 231, 312 and 321. (This pattern does not appear in cup
draws as there are an odd number of teams!)
Four balls in a bag is a classic semi-final draw, and I have covered
this in a previous post. Hopefully it is clear by the same logic that
there are 4 x 3 x 2 x 1 = 24 possibilities, and the table in the post covers
them all for the conspiracy theorists.
These numbers are known as factorial numbers and denoted by the ! symbol
in conventional mathematical notation. 1! = 1, 2! = 2, 3! = 6 and 4!
=24. So the first number that we need for our quarter-final analysis is
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320. In other words, the sequence
43718526 that I mentioned above is only one of 40320 unique possibilities for
the appearance of the balls.
(We will need to remember later that some of the different sequences
have the same sporting outcome in our particular example. If the teams
appear in the order 12345678 then the matches will be the same as if they
appear 78563412. More of this later.)
(These numbers get pretty large in the earlier rounds of the
competition. Try working out the number of possible draws for the third
round of the FA Cup, which has 64 teams! It's about one hundred and
twenty-seven thousand billion billion billion billion billion billion billion,
using a billion to mean a million million. Perhaps we'll come back to
this next January, if you are still speaking to me.)
Step
Two – What is the chance that Man Utd will play Chelsea?
In my question, I made no distinction about which team was home or away
on the first leg, so common sense takes us the next step. Assuming the
draw is fair, Chelsea is one of seven possible opponents for Manchester
United. All of these opponents are equally likely, so the chance of any
named one is 1 in 7 (expressed as betting odds), one seventh (expressed as a
fraction of one) or 14.2857% (expressed as an approximate percentage
chance). You can see this by working out 100 divided by 7 on a
calculator, and it is an example of a recurring infinite decimal. One
seventh is 0.142857142857142857…. with the six numbers repeating in the same
order.
Putting this visually, the size of this rectangle represents all the
possible 40320 draw outcomes. The reason why this is a 5x7 grid will
become clear shortly.
In one-seventh of these (5760 of them to be precise), Manchester United
will play Chelsea. In another 5760 of them, Manchester United will play
Real Madrid. Manchester United have to play someone, and 7 x 5760 = 40320
as all seven opponents are equally likely as we said.
So the green-shaded area represents the 5760 possible draws in which
Manchester United would play Chelsea, and the other six teams are playing each
other in all the various permutations.
Step
Three
What
is the chance that Man Utd will play Chelsea AND Real Madrid will play
Barcelona?
For the second part of my question, we have to focus ONLY on these 5760
possible draws that put Manchester United and Chelsea together. Real
Madrid may play Barcelona in lots of possible draws in which, say, Manchester
United played Bayern Munich and Chelsea played Dortmund. Those would be
irrelevant to the question, which is about the chance of two things happening
simultaneously. To mathematicians, the word AND is very different from
the word OR in these types of problem.
So let’s assume that Manchester United are to play Chelsea, in one of
those 5760 possibilities shaded in green. Real Madrid must be playing one
of five other opponents, equally likely. So one-fifth of those 5760
outcomes will have Man Utd v Chelsea AND Real Madrid v Barca. 5760
divided by 5 is 1152. In other words, 1152 of the 40320 possible draws
will have these two pairings. (The other two pairings could be either way
round, it wouldn’t affect the answer to our question.)
Visually …
The dark-blue segment represents our answer in which Man Utd play
Chelsea AND Real Madrid play Barca. It is one-thirtyfifth of the whole
rectangle.
So we have two ways to calculate the final answer.
The fraction one-thirtyfifth as a percentage is worked out by pressing
100 divided by 35 on a calculator and we see 2.85714%. Again, those
numbers would repeat in an infinite decimal. The same answer is achieved
by calculating the odds as 1152 divided by 43020 and multiplied by 100 to get a
percentage. 2.85714% again – it’s really exactly the same destination,
just reached by two different routes.
Step
Four – Finally, why does any real draw have a 0.059523809% chance of happening?
Remember that there are 40320 unique draws – each one is a unique
sequence of the eight balls. The chance of any individual sequence, say
14528763, is 1 in 40320 (expressed as odds), 1/40320 (as a fraction) and
0.00248015873% as a percentage. If you try 100 divided 40320 on your
calculator it will either round it to something like 0.00248 or will show it in
something called standard form as 2.48015873 x 10-3. I’m
guessing that if that means anything to you, you have easily understood the
rest of this post so I am moving on quickly. It’s a small chance – one in
about forty thousand.
However, in this particular context it doesn’t really matter to us which
order the four ties come out. Remember I said earlier that we would need
to come back to this point. With four ties to be drawn, there are 24
possible ways of getting the same sporting pairs with the same home/away order
(by the same logic as the semi final draws), so my final step is to say that
this combination of teams in the right home/away order would have come up from
24 out of the 40320 possible outcomes, and the same type of calculation gives
us 0.059523809%. 24 divided by 40320 multiplied by 100 if you want to check.
Remember, you will be able to say,
“Well, that had a 0.059523809% chance of happening!” EVERY year after EVERY FA
Cup Round Six draw. The numbers will
always be the same, and if that doesn’t win you gasps of admiration I’ll be
amazed.
Postscript
– What were the chances of an all-Spanish tie?
There are three Spanish teams in the draw, so there could be one all-Spanish
tie, or none. Going back to our visualisation, the green area represents
all the possible 5760 outcomes in which Real Madrid play Barcelona.
Atletico could be playing any one of the other five teams. The orange
area represents all of those (another 5760) in which Real play Atletico, and
Barcelona could be playing any one of the other five. The purple area is
for Barcelona playing Atletico.
We can see that as a fraction this is three-sevenths, or (3 x 5760) out
of 40320 = 17280 / 40320. Either route leads us to 42.8571% and those
same digits in a different order! Beautiful, I think you'll agree.
So
there we are. Welcome to my world. I told you there'd be tangents,
and remember, "Chance is a Fine Thing". :)
