Predict Euro Cup Matches with Simple Statistics: should I take that bet? (Part 3)
How to predict the quarter-finals
The dream
The dream is to win the friendly football poule
That is not my quote, but at this point, it might as well be. I also like math and statistics, so that sounds like a match. Let’s get into it.
Expected value
This is probably the most important concept when it comes to winning bets. It is the mean expected outcome of your bet, considering all available options and payouts. Let’s look at an example.
The coin toss
Someone presents you with a head-or-tails bet: you call head or tails, flip a coin and then one of the following happens:
— if you are correct, you get one dollar
— if you are wrong, you lose two dollars
Should you take this bet?
Answer: no (unless you like losing money). You probably knew that by instinct, but why exactly?
To find the exact answer, we have to find the likelihood and the payout of each option and take a weighted average.
Let’s say you called heads, assuming a fair coin, there is a 50% chance of the coin showing heads, meaning you would be correct. See also the table below.
+-------+---------+---------+
| event | chance | result |
+-------+---------+---------+
| Head | 50% | +1$ |
| Tails | 50% | -2$ |
+-------+---------+---------+The formula is SUM(chance * result) for each option. The expected value of the above bet isE = 0.5*1 + 0.5*-2 = -0.5. This means if you repeat the bet enough times, you lose $0.50 per bet on average.
Pig
Just one more example before we go straight to the football statistics, I promise.
There’s plenty of dice games where this concept applies, for this example we will look at Pig.
The rules are simple:
You repeatedly throw the die
— If it is a 1, you end the turn with no added score
— If it is any other number, it is added to your turn total and the turn continues
— A player may choose to “hold”, this adds the turn total to their score.
Example:
You throw a sequence 3–4–3–5, totalling 15 points. You hold and add 15 points to your score. Your opponent throws a 3 and then a 1, they add nothing to their score.
When should you hold?
When to hold depends on your opponent’s score and the win condition, but let’s just try to get the maximum average total in a single turn.
When you start your turn, the table is as follows:
+-------+---------+---------+
| event | chance | result |
+-------+---------+---------+
| 1 | 16.67% | -0 |
| 2 | 16.67% | +2 |
| 3 | 16.67% | +3 |
| 4 | 16.67% | +4 |
| 5 | 16.67% | +5 |
| 6 | 16.67% | +6 |
+-------+---------+---------+The expected value of your next throw is:
E = 0.1667*-0 + 0.1667*2 + 0.1667*3 + 0.1667*4 + 0.1667*5 + 0.1667*6
= 3.333That’s a positive number, meaning we expect to gain from this throw, which makes sense, it is our first throw.
Let’s say we rolled a 4, our next turn has the following table:
+-------+---------+---------+
| event | chance | result |
+-------+---------+---------+
| 1 | 16.67% | -4 |
| 2 | 16.67% | +2 |
| 3 | 16.67% | +3 |
| 4 | 16.67% | +4 |
| 5 | 16.67% | +5 |
| 6 | 16.67% | +6 |
+-------+---------+---------+We now have a turn total of 4 which we can lose when we roll a 1, the table is updated accordingly.
New expected value:
E = 0.1667*-4 + 0.1667*2 + 0.1667*3 + 0.1667*4 + 0.1667*5 + 0.1667*6
= 2.666You can see our expected value decreases. This makes sense: we can gain the same but we can lose more. So we can rephrase “when should we hold?“ to:
When does our E become negative?
For this we solve the following equation, note the X:
0 > 0.1667*X + 0.1667*2 + 0.1667*3 + 0.1667*4 + 0.1667*5 + 0.1667*6
0 > 0.1667*X + 3.333
-3.3333 > 0.1667*X
-20 > XThe strategy for Pig should be:
— if the turn total is lower than 20, throw again
— if the turn total is higher than 20, hold
Football
I promised, so here it is. Those of you who read Part 2 probably already know where I am going with this.
We will use the heatmaps to look at the chances of a match outcome occurring, and we will calculate the expected value for each possible bet.
If you do not know of these heatmaps, I recommend reading Part 2 first.
Poule scoring
My football poule has the following rules:
Full score correct: 10 points
Winner correct: 5 points
Tie correct: 7 points
An additional 2 points are awarded for each correct number of goals.
See also below for more examples on scoring for my poule.
To make your own calculations, update the values for whatever scoring system you use.
Upcoming matches
We will predict quarter-finals, the following matches will be played:
+-------------+-------------+----+----+--------------+
| Country 1 | Country 2 | r1 | r2 | group |
+-------------+-------------+----+----+--------------+
| Switzerland | Spain | 13 | 6 | small disadv |
| Belgium | Italy | 1 | 7 | small adv |
| Denmark | Czech | 10 | 40 | large adv |
| Ukraine | England | 24 | 4 | large disadv |
+-------------+-------------+----+----+--------------+This time, no home advantages, so we will use the Neutral heatmap.
I will show the calculations for the match Denmark — Czech.
Looking at the heatmap for “large advantage” we see 1–0 is the most likely outcome. Does this also yield the highest expected value? Below is the table with the associated odds and results if we predict 1–0.
Multiple match outcomes on one row indicate they result in the same payout, the chance is the sum of all outcomes.
+-----------------+---------+---------+
| event | chance | result |
+-----------------+---------+---------+
| 1-0 | 16% | +10 |
| 2-0 3-0 4-0 | 20% | +7 |
| 2-1 3-1 3-2 | 17% | +5 |
| 0-0 1-1 1-2 1-3 | 30% | +2 |
| 0-1 0-2 2-2 0-3 | 17% | +0 |
+-----------------+---------+---------+The expected value for this bet is:
E = 0.16*10 + 0.20*7 + 0.17*5 + 0.30*2 + 0.17*0 = 4.45Now we have to see if another bet might yield a higher expected value, I will choose the second most likely outcome: 1–1. See updated table below:
+-------------------------+---------+---------+
| event | chance | result |
+-------------------------+---------+---------+
| 1-1 | 14% | +10 |
| 0-0 2-2 | 15% | +7 |
| 0-1 1-0 1-2 2-1 1-3 3-1 | 44% | +2 |
| other scores | 27% | +0 |
+-------------------------+---------+---------+The expected value for this bet is:
E = 0.14*10 + 0.15*7 + 0.44*2 + 0.27 * 0 = 3.33This is interesting: even though the chance of it being entirely correct is only a bit smaller, 14% vs 16%, the expected value is much lower, 3.33 vs 4.45.
It is clear the expected value depends on what number of points are awarded for each type of bet. In my poule, ties are simply not very attractive bets.
The predictions
Below is the table with the predictions and the expected value of each.
+-------------+-------------+--------------+-------+-------+
| Country 1 | Country 2 | group | PRED | E |
+-------------+-------------+--------------+-------+-------+
| Switzerland | Spain | small disadv | 0-1 | 3.25 |
| Belgium | Italy | small adv | 1-0 | 3.43 |
| Denmark | Czech | large adv | 1-0 | 4.45 |
| Ukraine | England | large disadv | 0-1 | 3.97 |
+-------------+-------------+--------------+-------+-------+Important note about the crystal ball
I do not own one, I simply aim to make statistics a bit more fun. Any risks you take are your own.
The end
Thank you for reading.
If you play with a poule with a different scoring system, feel free to leave a comment about how it works.
Here are my links:
Read more
This story has a Part 4, Predict Euro Cup Matches with Simple Statistics: how to rise the ranks (pt. 4).
More about the scores
— Entire score correct: 10 points
Example:
predicted 2–0, outcome 2–0
— Winner correct, number of goals correct for 1 team: 7 points
Example:
predicted 2–1, outcome 2–0
— Tie correct, number of goals incorrect: 7 points
Example:
predicted 1–1, outcome 2–2
— Winner correct, number of goals wrong for both teams: 5 points
Example:
predicted 2–1, outcome 1–0
— Winner wrong, number of goals correct for 1 team: 2 points
Example:
predicted 2–1, outcome 0–1
