This weekend I thought about the flashlight and 8 batteries riddle:
You have a flashlight that works with 2 batteries. You also have 8 batteries, of which 4 are empty and the other 4 are full. There is no way to tell which battery is empty and which one is full, but you can put 2 batteries in the flashlights. If both of them are full, the flashlight will turn on.
What is the minimum number of tries that will guarantee that the flashlight will turn on?
At first, I explored the possible combinations, then I considered playing with probability trees. But then I put this on paper, and soon, things became much clearer. I represented the batteries with dots, and the tries with lines, and instead of playing with abstract concepts, I started to play with lines and dots:
First I found a solution that would try 7 combinations, and would guarantee that the 8th was correct:
But somehow I knew that the solution had to be 7 tries, not 8. So I kept moving the lines and connecting the dots until the bulb lit (pun intended):
No matter where the 8th full battery is, there will be a line connecting it with another full battery. (You can also find a video here).
This shows the importance of visualizing your problem before being able to come up with the answer. I am using a Moleskine notebook and a Baron Fig Squire pen (some say it’s the best pen in the world 🙂 ).
This is a follow up to https://colorblindprogramming.com/round-probabilities-before. Last year I stopped after discovering that the only correct way to calculate the odds is to look at the probability trees. This year I took this one step forward and created a script that would calculate the correct probabilities. I intend to reuse this script for the future draws, and a year it’s a long time for my memory so I am adding some notes here.
The incorrect approach: the big-bowl
The first approach last year was to calculate all the possible pairs, eliminate the invalid ones and then calculate the associated percentages for each pair. In hindsight, this approach was obviously wrong, because it doesn’t replicate the actual draw. This approach would only be accurate if the draw consisted of a single draw – from a very big bowl of all the valid options. This is obviously not how the actual draw works, so even if the final numbers were pretty close to the correct ones, it was not the correct approach.
The correct approach, using conditional probabilities
The correct way to look at this is by understanding that we are talking about dependent events. Each draw depends on the actual result of the previous draw. It’s identical to this process, beautifully explained on MathIsFun.com:
So how do we actually do it?
There are two approaches: The first one is a bit more complicated and implies creating the tree above for the 16 teams and 16 steps (each team pick is a step). It has the advantage of producing accurate results, but it’s a bit more difficult to implement. The second one consists of simulating the draw process and repeating it a lot of times. I found this approach easier, here is the pseudo-code of the draw process:
for each unseeded team
if there is a mandatory draw (starting from the 5th unseeded team)
then automatically create the pair and add it to the draw
otherwise, pick a random unseeded team
get the list of available seeded teams
randomly pick a seeded team from the list above
add pair to the draw
Repeating this process a few millions of times would lead to millions of possible draws, and based on that we can calculate the percentages.
But there are 2 catches: 1. Checking both sides of the draw. Have a look at the step 2 above, checking if there is a mandatory draw: let’s say you are left with 4 unseeded teams and 4 seeded teams. It’s not enough to look at the unseeded teams options, you also need to look the other way around. Example: Unseeded teams: Liverpool, United, Shalke, Lyon Seeded teams: PSG, City, Real, Barcelona Liverpool has 2 options, United 3, Shalke 4 and Lyon 2. But if you randomly pick Shalke and you pair it with any of PSG, Real or Barcelona, then you leave an impossible draw for City (which cannot be drawn against any of the 3 English teams left). So the solution is to count the number of options for both unseeded and seeded teams. If there is a single option, pick it.
2. Go back if needed. Even with the above safety mechanism in place things can still go wrong. Example: Unseeded teams: Roma, Liverpool, Shalke, Lyon Seeded teams: Porto, Barcelona, PSG, City Options for the unseeded teams: Rome -4, Liverpool -2, Shalke -4, Lyon -2. Options for the seeded teams: Porto -3, Barcelona -4, PSG -2, City -2. The safety mechanism above (counting the number of options for both seeded and unseeded teams) tells us that everything is fine. So we go ahead and pair Rome with Porto. We are now left with: Unseeded: Liverpool -1, Shalke -3, Lyon -1 Seeded: Barcelona -3, PSG -1, City -1. The problem is that both PSG and City have an option, and that option is Shalke. So this leads to an impossible draw, so the solution in this case is to go back one step and pick another draw instead of Roma v Porto. According to my calculations this could happen in about 0.4% of cases, and I am really curious how UEFA would handle it if it happened on stage. In the scenario above, if Roma was selected as unseeded team, I expect that the computer will only allow PSG and City to be one of the seeded teams, but I am really curious to hear the hosts explanation about this constraint (since both Porto and Barcelona are, at first sight, also valid options for Roma) 🙂
Using the algorithm above, I ran the simulation 2 million times. These are the results:
Checking the results
The nice thing about being both a geek and a football lover is that you get to know smart persons at the intersection of science and football. Two of them are Julien Guyon and Emmanuel Syrmoudis. They also spent time thinking about this topic. Julien came up with a great explanation of the draw process and probabilities, while Emmanuel went one step forward and actually created an interactive draw simulator.
My results come pretty close to theirs, so I’m quite confident that my method is decent enough. I plan to reuse it again next year and, perhaps, also try to create the actual probability tree to get the exact percentages.
I wrote a piece about football analytics in Romanian: when football meets science. It was one of the articles I really enjoyed writing and it took me over 10 evenings to do it.
Here are the top level details:
Football analytics is all about using data about previous events in order to have an indication about the outcome of future events.
It is not new: it started somewhere in the ’50 and one of the first coaches to use it was a Russian trainer called Valeri Lobanovsky, in an era where a computer was taking up rooms.
I found a correlation about the DIKW pyramid and the usage of football data: – Data – numbers and metadata collected using manual operators, tracking devices or video tools – Information – when data is put into context. One indicator that recently became mainstream is the ‘expected goal‘ (xG) – a percentage associated with every shot based on previously aggregated data – Knowledge – when information is combined with previous experience. Example – aggregating information about indicators like xG (xG for, xG against, non-shot xG, xG difference) – Wisdom – using previous levels to take strategic decision enabling competitive advantage.
The first two levels are for the football fans, media writers and TV pundits.
The last two levels are for the professional football clubs and for the betting companies. This is where the football analytics takes places and these levels can give indication about future events.
A few examples of football analytics: 1. transfers: before any transfer, the targeted player is analysed from a few perspectives: tactical, physical, technical. The modern clubs are using players databases with custom criteria in order to maximize their match rate. 2. injury prevention: by tracking the way a player runs and measuring how long his feet stays on the ground, one can evaluate the player tiredness 3. predicting outcome of future events by calculating and maintaining a club index (ex. fivethirtyeight.com) 4. penalty shoot-out: statistics showed that the team shooting first has a 20% advantage over the second team. The football governing bodies realized this un-fair advantage and recently changed the order of the shoot-out (now ABBA instead of ABAB)
In the end, football remains a random sport. Using analytics can give indications, and make the clubs better understand some questions, but it cannot (yet) give definite answers. As long as football is played by humans, the human factor will play its part and will keep football random and enjoyable.