Can anybody remember being in the maths lesson at school and asking what’s the point of learning about angles and distributions as you never use them in real life.  Well in football, the Poisson distribution is the key to the number of goals and hence the drive for points awarded for fantasy points (goals and assists).
Observing 11,000 matches (so 22,000 team scores), scoring an average of 1.34 goals each have:
<img loading="lazy" decoding="async" class="alignnone size-full wp-image-1443" src="https://cms.fantasyfootballhub.co.uk/wp-content/uploads/2017/09/chart-1.png" alt="" width="661" height="405" />
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When I first read about goals following a Poisson distribution, I thought maybe the high scoring figures would be due to top teams v bottom teams.  After collecting data, on teams who had expected to score 0.9 (1,400 observations), 1.3 (2,500 observations) and 1.7 (1,000) we produce:
<img loading="lazy" decoding="async" class="alignnone wp-image-1444" src="https://cms.fantasyfootballhub.co.uk/wp-content/uploads/2017/09/chart-2.png" alt="" width="629" height="448" srcset="https://cms.fantasyfootballhub.co.uk/wp-content/uploads/2017/09/chart-2.png 564w, https://cms.fantasyfootballhub.co.uk/wp-content/uploads/2017/09/chart-2-300x214.png 300w" sizes="(max-width: 629px) 100vw, 629px" />
These probabilities very closely match to bookies prices.  However, it is worth noting that probabilities of home and away goals are not independent, and need adjusting if using for betting.
There are spread-betting companies give the expected number of goals a team will score in a match.
<h3>How does this help in selecting my team/captain?</h3>
Most fantasy football games give points for a clean sheet and minus figures for goals conceded.  We can use the probabilities from the Poisson distribution for the number of goals conceded with the points awarded for that number of goals to create an expected score.  For example, at the weekend Crystal Palace had an expected number of goals to score at Burnley as 1.025.  I had a Burnley defender so his expected return from defending was:
<img loading="lazy" decoding="async" class="alignnone size-full wp-image-1445" src="https://cms.fantasyfootballhub.co.uk/wp-content/uploads/2017/09/table-3.png" alt="" width="552" height="112" srcset="https://cms.fantasyfootballhub.co.uk/wp-content/uploads/2017/09/table-3.png 552w, https://cms.fantasyfootballhub.co.uk/wp-content/uploads/2017/09/table-3-300x61.png 300w" sizes="(max-width: 552px) 100vw, 552px" />
It turned out that Burnley had a clean sheet so I received more than expected.  This is being on the right side of variance.  Crystal Palace missed a hat-full of clear chances and on another day it would have been a negative score.
For attacking points, this is a bit more straight-forward as you really only need the expected number of goals a team will score and multiply this by points for a goal and the percentage of goals the player scores for a team and the points per goal (and similarly for assists).
This week I had Kane playing upfront.  Spurs were expected to score 1.475 goals, based on last season I estimate Kane scores around 37.9% of Spurs goals when playing.  So his expected goal return is 1.475 x 0.379 x 5 (pts for a goal) = 2.8 and for assists 1.475 x 0.091 x 3 = 0.4 giving a total attacking return of 3.2
We can now compare across player to select the highest point scorers for captain, but also in the official fantasy game determine which 11 should be selected for the game week.
<h3>How many goals/assists?</h3>
The one unknown of the calculations above, is the percentage of goals and assists a player scores when their team scores. One option is to divide total goals a player scored by the number of goals their team scored last season.  This would under-estimate percentages for players that were injured (eg. Kane only started 29 matches) or players signed in January (Jesus for Man City).
Ideally, you’d look at just matches a player played in and calculate the percentages.  This would be a very time consuming process with over 500 players and information not being really available.
To estimate these percentages, for each player who made more than 10 starts, I’ve multiplied their goals (and assists) by 38 games and divided by (their number of starts plus sub appearances x 0.5).  I think this is a reasonable estimate, though there could be scenarios where this could give wrong estimates.
For players that have made less than 10 starts last season, I assume they would take the average for that position or the same as another player within the team (eg. Jesus given same <a href="https://cms.fantasyfootballhub.co.uk/opta/" data-internallinksmanager029f6b8e52c="1" title="FPL Stats by OPTA - Player and Team Stats">stats</a> as Aguero, Salah same as Mane).
<h3>Now have a coffee and relax</h3>
Now relax with your coffee, as most of these calculations can be done within a spreadsheet and adapted to various fantasy football games.  The link to the spreadsheet below is a version that works for Sky and the official game
<a href="https://1drv.ms/x/s!AkYcP_eLvYMVjh2hVq0ridhJUA9v">Spreadsheet link</a>
There are currently only 6 pages, so I will give a brief description of each.
Games – This contains the upcoming matches and the expected goals derived from spread companies.  To the right is the distribution of goals scored for each team.
Sky teams – I find it slightly annoying typing in my team into a spreadsheet.  So I have created a table in this page for you to enter you team and that can be edited.  This is in the table (C23:Q34).  You enter your team name in the first row and the players in the cells below.  Note these have to be in the exact format as given by Sky.
I don’t know the formats, so I have created a search function.  In cell H4 you can select a position from the drop-down menu and the team from H5.  A list of players matching the criteria appears in column J.  You can now copy the players name and paste the value into the table.
Finally, in cell C2, select your team name and your team will appear in cells (C4:C14) which will be used in the next page.
Sky – Row 26 down-wards gives for each player (that has started at least 10 matches last season) the average number of bonus points per match plus the expected percentages of goals/assists for each goal the team scores.  If a player is coloured green, then I have assumed their <a href="https://cms.fantasyfootballhub.co.uk/opta/" data-internallinksmanager029f6b8e52c="1" title="FPL Stats by OPTA - Player and Team Stats">stats</a> are the same as the player above him.
The team selected from the previous page is displayed in B5:D15. The average points awarded for each type of bonus and for cards given per match is in columns F:J. Column K gives the number of points for a clean sheet and column L calculates the expected defensive points awarded given the Poisson distribution of goals scored by the opposition.  Finally, column M is the expected points your player is expected for each goal his team scores with column N giving the total number of points the player is expect for get via goals scored in the upcoming match.
Column O adds up the various columns to give a total of expected points (though excludes MotM and appearance points).  This should guide you in your captain choice (eg selecting the highest expected point scorer).
FPL teams – This follows the same structure of the “Sky teams” page.  Please note that here you need enter the goalkeepers in the green section of the box.  Also, the search function is based on Sky positions, let me know of any differences with the official games and I’ll update accordingly (already done this for Alexander-Arnold, T).
FPL – Again follows a similar format to the “Sky” page in terms of structure.  The expected points per player are given in column K.  This excludes appearance points and any bonus points.
The official game is choosing 11 from your squad of 15.  Based on the expected points of each player in the squad, cells C21:G32 gives the best formation and starting 11 with a guide to best option to captain.
Est gl assist % &#8211; This page contains the Sky <a href="https://cms.fantasyfootballhub.co.uk/opta/" data-internallinksmanager029f6b8e52c="1" title="FPL Stats by OPTA - Player and Team Stats">stats</a> from last season used to make the various calculations.  It is kept here for transparency and more detail if the user is interested.
<h3>Next steps</h3>
I hope this helps you in team and captain selection.  If there are some tweaks you would like introduced then please discuss in the comments section.
I plan to update the matches with expected goals at least the evening before the first game match.  However, this does depend on external factors (social life and information being available).

Having a coffee and a Poisson to select team/captain

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