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!

See also:

• https://www.reddit.com/r/LiverpoolFC/comments/1j2vj0j/when_can_liverpool_win_the_premier_league/
• https://www.reddit.com/r/LiverpoolFC/comments/1j87tzr/when_can_liverpool_win_the_premier_league_update/

Update after GW 29

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:

Here are some useful (and sometimes unconventional) travel tips from Kevin Kelly:

• Laser out, meander back. When you arrive in a new country, immediately proceed to the farthest, most remote, most distant place you intend to reach during the trip. Do not rest overnight in the arrival city. Then once you reach your furthest point, unpack, explore, and work your way slowly back to the big city, wherever your international departure airport is.
• Paradoxically, the best way to avoid that is to give strangers your trust and treat them well. Being good to them brings out their good. If you are on your best behavior, they will be on their best behavior. And vice versa. To stay safe, smile. Be humble and minimize your ego. 
• Take a chance. If things fall apart, your vacation has just turned into an adventure. Perfection is for watches. Trips should be imperfect. There are no stories if nothing goes amiss. 
• Counterintuitively, the longer your trip, the less stuff you should haul. Travelers still happy on a 6-week trip will only have carry-on luggage. Use laundry detergent sheets if needed and pack a layer no matter what.
• When you are traveling you should minimize the amount of time you spend in transit—once you arrive. The hard-to-accept truth is that it is far better to spend more time in a few places than a little time in a bunch of places. 
• A souvenir should have some meaning from the trip. When buying a souvenir, ask yourself where will it live once you get home. The best souvenirs from a trip are your memories of the trip so find a way to commit them to your memory: keep a journal, send updates to a friend, make a photo book. 
• When asking someone for a restaurant recommendation, don’t ask them where is a good place you should eat; ask them where they eat. Also, walk at least five blocks away from a tourist attraction to get an inexpensive and authentic meal.
• The list of most coveted cities to visit have one striking thing in common—they are pedestrian centric. They reward walking.
• When checking the status of your flight, first check whether your plane has even arrived at your departure airport. Put AirTags into your bags, so you can track them when they are out of your sight. 
• The Google Translate app for your phone is seriously good, and free.
• Organize your travel around passions instead of destinations.