The Probability Thread 2009

Poundstretcher

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So, it's that time of the season again. The time when we start to get our calculators out and we begin to wonder about chance, probability and stochastic independence (well, that last one might just be me). I figured that it wouldn't be long before Lammie started nagging me about numbers - and in fairness, the "how many points..." threads have been dying for a definitive contribution for some time - so I settled down to figure out the best way of approaching things.

We've got an unbelievably tight division this year - with 78 games to play, every last fixture impacts on something: form and fortune interplay so closely that we could conceivably finish almost anywhere in the top half, and the playoff candidates could form up from any of many. So the logical way was to run a Monte Carlo experiment: simulate every match from here until the first weekend of May and sum the league table up to that point, before taking into account its implications for future fixtures. There are, in fact, 16,423,203,268,260,658,146,231,467,800,709,255,289 different result combinations from here to the end of the season (that's 16 billion billion billion billion) - even without thinking of goal difference...

I calculated probabilities in the Bayesian manner, looking to use home and away records separately, and averaging out the win/loss percentages to find the chance of particular outcomes. I then used a random process to simulate the result, randomly selecting the scoreline using a proportional representation of the season's league two results to date, and then updated the league table (e.g. 24% of home wins are 1-0, 16% are 2-0, etc).

For example, the next game of the season is Accrington v Lincoln. Stanley at home have an 8-5-7 record (40%-25%-35%), while the Imps are 7-3-9 on the road (37%-16%-47%). Averaging these out gets you a 44% chance of a home win, 20% on the draw and 36% on the away win. With the scoreline, this data was added to the league table, and used for the subsequent matches.

I then ran the entire league programme through 500 times (that's 39,000 simulated matches), to find out: (a) the average number of points gained by the third-placed side, (b) the average number of points gained by the seventh-placed side, (c) City's finishing position, and (d) the identity of the Champions.

Points for Promotion
This graph seems to bear out much of what's been discussed - 78 points is pretty certain to get you there, but if you don't get that many, then you've still got a chance. Four wins from our last six could well see us up. Mean: 76.16



Points for the Playoffs
It's conceivable that one more point could make the playoffs... conceivable, but really not likely. 72 points will get you there beyond reasonable doubt - two wins from six should mean we make the playoffs: should be no last-day worries here... in theory. Mean: 70.06



City's finishing position
Saying that, there's still a 15% chance we'll miss the playoffs altogether... and finishing so poorly we end up 12th isn't out the question. Still, at the other end, we have better than a 1 in 4 chance of making the top three - and one year in every 45, we'll be Champions from here. Mean: 5.13



Who else could win it?
The numbers point very, very convincingly towards Brentford. Don't rule the others out - anything can happen, after all, but three times in four the title goes back to West London.



Oh, and MikeB, if you're reading this, feel free to stick this in the programme. Everyone should have the chance to read mathematics this interesting.
 

International

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I assume you got have had your A-Level students submit their coursework again :) I (let alone the probability of LLL asking) was only about to start a thread suggesting that it is this time of year that you supply us with such mind boggling statistics.

Mrs Int asks if you are single? I think she now has more than a passing interest in football when you present it like this
Too late to edit
 

Oli

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What would be interesting is to run it at the half way stage and then compare it to the odds at the bookies and see if you would actually make any money.

A lot more work, I have to admit.
 

CWH-NYC_

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Great stuff-but I am sure knowing The City it will all come down to a last day roll-a-coaster of epic proportions...start drinking heavily Now!
 

Ashecfc

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Thats very impressive, but unfortunatley i think, your "model" does not include suspentions, injurys, moral and other psycological affects, and as such will reduce the overall accuracy of the results.

My conclusion is the same as CWH-NYC and it will come down to last game of the season, I do realise that your "model" gives the end result and not a time-line a such.

Anyway bloody good work!! [y] and well done

and as Oli says with the odds from the bookies you could find some value for money!
 
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