Over 2500 kilometers through France, Switzerland and (a touch of) Italy.
New record with a single charge: 324km in Italy. Not planned, but forced by an incident next to an intermediary supercharger. Supposed to arrive with only 7% battery, made with 9% thanks to 90kmh driving in the trucks lane.
Impressed by the regenerative energy feature: during the Switzerland to Italy pass, I was down to 7% battery and the system was telling me I will arrive with 8-10%. I arrived with 14%, but not looking forward to do it again.
So the battery is still going strong.
Can’t say the same about the AC compressor, which failed a couple of weeks before the road trip. Fortunately I was able to replace it on time.
2500kms round tripNew record between Aosta and ImperiaIn Southern France
With 9 games to go, Liverpool are top of the Premier League with a 15 points advantage:
Arsenal now have less than 1% chances, and the bookmakers offer 41 to 1 odds if you bet on Arsenal. It’s been a bit of a rollercoaster though, just watch the fall from 3.75 on 19 February to 26 just one week later:
Liverpool navigated very well the fiery February, but, with two draws that felt like two losses (at Everton and Villa), they needed Arsenal to drop some points. Arsenal dropped more than expected, and now it’s simply a matter of time until Liverpool are crowned PL champions.
I wanted to find a scientific way to determine when this could happen. And the answer is this:
How to read it?
The chart says that the earliest that Liverpool can win the PL is in match day 32 (with a 15.3% chance), when they will play West Ham at home. The most likely is the next match day (Leicester away, 35.9%), and by the time they play Arsenal (MD 36), there is a 90% chance that the PL title is already won. Arsenal only have a 0.35% chance to win the PL (not shown in the chart).
How does it work?
I used the xG data from fbref.com, averaging the xG at home and away for every PL team. If we take the next Liverpool game, against Everton: Liverpool have a 2.18 average xG at home, while Everton have a 0.99 average xG away. I used a Poisson distribution function to turn these xG numbers into actual goals, and, ultimately, into winning percentages: 65% Liverpool win, 19% draw, 16% Everton. This is how it looks like:
I ran 100.000 simulations of the remaining PL games for both Liverpool and Arsenal and I came up with the percentages in the chart above. Interesting fact: running 10.000 simulations on an Intel Core i5 8th generation took about one minute. The same number of simulations on a MacBook M1 Pro took 13 seconds; so I could afford to run 100.000 simulations. However, running 10x more simulations did not change the data more than a rounding error, so for all intents and purposes, 10.000 simulations are sufficient.
More data
10 PL teams can no longer catch Liverpool. 2 more can follow if Liverpool win their next game against Everton at home, and another one (Bournemouth) if they don’t win their next game:
According to my simulations, 80 points are sufficient to win the PL in more than 80% of cases. So 10 more to go!
In the Seinfeld episode ‘The Kiss Hello’ (S06E17), Jerry’s Nana remembers that Uncle Leo was supposed to give Jerry’s mom 50 dollars 53 years before. When Morty (Jerry’s father) finds out, he starts to calculate the amount that Uncle Leo would owe his wife, Helen:
MORTY: Do you know what the interest on that fifty dollars comes to over fifty-three years?
HELEN: Oh, Morty, please.
MORTY: Six hundred and sixty-three dollars and forty-five cents. And that’s figuring conservatively at five percent interest, over fifty-three years, compounded quarterly. #
Using the 5% interest rate that Morty mentions, the best case scenario for the $50 over 53 years would be for continuous compounding:
Total = P * e ^ rt, where P=50, r=0.05 and t=53, so Total ~= $707
The value that Morty came up with – $663.45 is extremely close to what he would have received for a yearly compound:
Total = P * ( 1 + r/n) ^ nt, where P=50, r=0.05, t=53 and n=1, so Total ~= $663.75
However he does mention “compounded quarterly“, so the final expected amount should be:
Total = P * ( 1 + r/n) ^ nt, where P=50, r=0.05, t=53 and n=4, so Total ~= $696.17
In conclusion, Morty underestimated the amount that Helen should have received by $32.42. Quarterly compound is better than yearly compound, and the continuous compound is the best. Also, $696.17 in 1995 (when the episode aired) is worth $1,451.14 today.
Intuition about the e number
Imagine an initial investment of $1, earning 100% annual interest. So P=1, r=1, t=1.
If interest is compounded once per year, you end up with: (1+1)^1 = 2
If interest is compounded twice per year, you end up with (1 + 1/2) ^2 = 2.25
Quarterly compounding: (1 + 1/4) ^ 4 = 2.44
Monthly compounding: (1 + 1/12) ^ 12 = 2.61
Weekly compounding: (1 + 1/52) ^ 52 = 2.69
Continuous compounding: e = 2.718
Discrete vs continuous growth
At its core, e measures the idea of continuous growth, which is different from the discrete growth. Discrete growth means that a given population only grows at certain moments (example at the hour mark). Continuous growth is different: instead of waiting until the end of the hour, the population is growing all the time, at every moment, like a plant that grows gradually instead of in sudden jumps.
The discrete growth is modelled by the (1 + r)^t formula, while the continuous growth is modeled by the e^rt formula. The second one results in a slightly higher number because you’re adding a little bit of growth continuously.
Assuming a population of 1000 astrophage, and a growth rate of 5% per hour, here is how it will grow over a 10-hours period:
Continuous growth is slightly faster than the discrete one. Image generated by the o3-mini OpenAI model.
