As a Blackburn Rovers season ticket holder I can’t believe that it’s come down to the last game of the season. By 5.45pm on Sunday 22nd May 2011, two of five teams currently separated by only a single point will have been relegated from the Premier league (Table 1). Comments from mates like “you’ll be safe” or “you’re too good to go down” don”t help when it comes down to a single game that can make or break your season. So what are the chances of Blackburn – or indeed any of the bottom teams – joining West Ham by going down on Sunday?

### What every fan wants to know is, what are the permutations that affect my team, and what is the probability of them going down?

The first key thing is that the game between Wolves and Blackburn affects all others – if either of these teams win then they automatically stay up. What is also pretty obvious is that if Wigan or Blackpool lose, then they automatically get relegated. But there are a lot of other combinations.

### How to predict who will get relegated

There are 3 scenarios in the game involving Wolves and Blackburn, i.e. a Wolves win, a draw or a Blackburn win. For each of these options there is the possibility of a Birmingham win, draw or loss making 9 scenarios in all. Similarly, for each of these 9 there are 3 in the Blackpool game – making 27 – and for each of the 27 there are 3 Wigan options. This makes for 81 scenarios in all (3x3x3x3) shown in Table 2 which can be used to see who would be relegated in any specific scenario (with the assumption that goal difference remains the same).

Figure 1 shows the number of senarios in which each team is relegated. For Blackburn, there are 9 scenarios out of 81 (11%) where they get relegated while for Wolves there are 13 (16%). It gets worse for the bottom three with Blackpool being relegated in around half of all scenarios and Wigan two thirds.

### Bizarre scenario no. 55

What would be the bizarest scenario we might see on Sunday? If Wolves and Blackburn draw and all the others win, then they will both get relegated (scenario no. 28). The one that appears to be the harshest, however, is scenario no. 55. In this case, Blackburn loses and all other teams win. Blackburn go down with 40 points and Wigan with 42 points. The only way out for Wigan in this scenario is for them to score lots of goals so that one of the others go down instead. For Blackburn, there is no way out.

Am I happier now? Well, a little. My team, Blackburn, is not relegated in 72 of the 81 scenarios. However, that does leave the other 9 where they are. There’s only one thing for it. At 4pm on Sunday 22nd May 2011, I will be going for a run that lasts around 2 hours.

### Update 20th May 2011

Given the number of comments who wanted to know about goal difference and how this would affect each teams chances, please see the new article on Survival Sunday in the Premier League: how to escape relegation.

ScaveedTrue, there are that amount of scenarios, but the question is, which are more likely? Given our (Blackburn’s) current form, and manager, we may greatly struggle against a determained Wolvees, and also a draw is unlkely in any game, so the pie chart does not accurately reflect our, or anyone else’s, survival hopes. The odds, for example, of Rovers going down are not 9 in 81.

Also, why is scenario 55 any harsher than scenario 1?

stevehaakeI’ve been trying to work out the best way to talk about chances and you’ve made a good point. For Rovers, if we win we have 100% chance of staying up. If we draw, there is only one scenario (55) where we go down. If we lose there are 8 scenarios where we could go down and we could say that we blew it. Not quite sure how to put this is % terms or odds on in number of scenarios.

One thing in Rovers’ favour is our goal difference which takes away quite a few of the scenarios.

I think scenario 55 is harsh, but on Wigan rather than Rovers. Can you imagine winning on the last day, getting 42 points and STILL getting relegated!

Thanks for the comment.

Savorythink of the combinations of bets you could make to soften the blow of us (rovers) goin down..

stevehaakeMight be a better way of spending Sunday?

Nick RichAmazing to know there are 81 scenarios without even throwing goal difference in!

Unfortunately that’s also a pretty big flaw as only the draw results would keep this as-is; actually, that’d also probably only be strictly true of a single scenario – where all the teams draw. If that doesn’t happen, at least one team will have different GD in any other scenario of the 81.

What would it look like if a reasonable scoring margin was included – say up to 3 goals – for win/loss? I’m assuming there’d be a whole heap more due to the additional series, but what effect there would be on the outcome is probably the interesting bit – such as the ratio of potential match outcomes to the potential match outcomes that would make a difference, e.g. would blackburn always stay up with a win whatever the win/loss margins in the games?

stevehaakeNick,

You asked the question I’ve been trying to avoid! The only way I reckoned to do this was by responding to individual questions; I tried to do it in a spreadsheet but my head exploded.

To answer your last question first, Blackburn and Wolves would always stay up if they win. But goal difference could have an effect on the losing team as follows:

The goal difference between Blackburn and the Birmingham, Blackpool and Wigan is 6, 7 and 8 respectively. So if Rovers lose badly to Wolves, 3-0 say, then Birmingham, Blackpool and Wigan would have to win 3-0, 4-0 or 5-0, respectively, to overtake them. This increases Rovers’ relegation scenarios by 3 for Birmingham, 6 for Blackpool and 7 for Wigan.

A similar thing happens for Wolves if they lose although the goal margins are a lot smaller. If Wolves lose only 1-0 to Rovers, then Birmingham, Blackpool and Wigan would only have to win 1-0, 2-0 or 3-0, respectively, to overtake them. This increases Wolves relegation scenarios by 4 for Birmingham, 5 for Blackpool and 8 for Wigan.

We could look at all the permutations for Birmingham, Blackpool and Wigan alone but things are so close that it is all pretty similar.

I haven’t looked at what happens when goal difference is the same – I presume it’s on goals scored?

Hope that helps rather than bamboozles!

Thanks for the comment.

Steve

Savorywell ive put a few quid on us losing & to draw and the others to win, so if we do go down, at least i’ll be a few quid up to help finance the amount of drink i will need to deal with it😉

stevehaakeA win-win situation (or is that lose-lose?)…

BenI don’t think you are quite right in your goal difference scenario – it affects Birmingham, Blackpool and Wigan if one or more of them lose or win, but involes Wolves and Blackburn if they lose and others draw.

Goal difference has the biggest effect on scenario 14 and others close to that number. If Blackburn beat Wolves by one goal (the minimum difference possible) and Birmingham draw then Birmingham now have the same goal difference as Wolves, and Wolves may or may not go down (depending on goals scored or whatever). If the difference is two goals then Blackpool can draw and have the same difference (Birmingham stay up), if three goals then Wigan have the same difference, but get relegated on 40 points as well as Wolves if Birmingham and Wigan also draw or win. MY HEAD HURTS!

Blackburn’s better goal difference is a real boost if it is they rather than Wolves that lose as then they would have to lose by 5 more goals to be in the same situation (i.e. 6:0, 7:0, 8:0) – surely not even Blackburn can manage that…

BenMore on the goal difference issue – a quick count says that about 30 of your 81 scenarios involve at least two teams drawing for 17th and 18th places. Goal difference is important for all these except for 41,42,44,50, where the teams in 17th and 18th have both drawn so goal difference doesn’t change.

This means that 26 scenarios have at least 2 permutations based on goal difference, some more (if three teams are drawn for 17th, 18th and 19th), so the possibilities are well over 100.

stevehaakeBen, even I’m starting to lose it now. You are right, scenario 14 is one to watch out for when it comes to goal difference. But good to know that there are at least 100 scenarios now!

BenFollowing Nick’s suggestion I ran a Monte-Carlo simulation (in Excel) to simulate all game results with each team in each game having an equal chance of scoring 0 to 3 goals. This is a big assumption as it doesn’t take into account the actual probability of each result – e.g. Blackpool’s chance of winning 3:0 at Old Trafford are probably quite slim.

Ignoring these issues I ran the simulation 100000 times and ranked the teams according to points, then goal difference then goals scored.

There were no absolute draws (equal points, goal-difference, goals scored).

The percentage times each team was relegated are given in the second column, with the odds from a well-known bookmaker converted to a percentage and given in the third.

Blackburn 13% 8%

Wolves 25% 11%

Birmingham 47% 62%

Blackpool 53% 58%

Wigan 62% 75%

Sum 200% 214%

The sum comes to 200% as two teams will be relegated. The bookies’ odds come to more because of their profit margin. On the face of it the bookies seem to underestimate Blackburn’s chances of relegation and quite severely underestimate Wolves chances. I think that just shows the limitations of the assumptions I made earlier, so don’t put your mortgage on it.

stevehaakeFantastic! The order is the same as my simple calculations here but as you say Rovers and Wolves go up a little in probability while the bottom 3 go down in probability. Shows that a first order approximation is a step change from knowing nothing and then a bit of thinking is needed to get an accurate result.

Are you saying that I ought to put a bet on Blackburn and Wolves going down? I suppose that might sweeten the pill.

Thanks for that.

(I must have a look at that Monte Carlo Simulation routine!)

damianone thing that you have not taken in to consideration and from what i can see is ommited from your various scenarios is you say if wigan and blackpool both lose they are relegated – look again this is not correct

if all 3 birmingham,blackpool and wigan all lose any 2 from three can go down

ie wigan lose 1-0 gd – 23, if birmingham lose 3-0 they are down they also would be on -23 but have scored less goals – if blackpool lose by less than 3 goals they are safe they would also be on -23 but have scored more goals than both birmingham and wigan.

i think there are 9 different permutations here to add to yours

bottom line is all 3 can lose and any of them can stay up dependant on the goals against.

stevehaakeDamian,

Initially, I took the approach that goal difference remained the same to keep it simple and to stop my head exploding. As you point out, as soon as you think about goal difference the bottom three all start to swap around. Ben (two comments above) did a new analysis where he took all the different scores that could happen into account (up to a maximum of 3 goals per team). He created 10,000 random scores for all the teams to work out what would happen. He also took into account goals scored in the case of a tie.

The new % is below (with equivalent scenarios out of 81 is to allow comparision with the original figures). Rovers and the Wolves chances of relegation go up while the bottom three’s chances improve. Goal difference does make a difference, especially so it seems to Wolves.

So, to answer your comment – you are right!

Blackburn 13% (=11 scenarios out of 81)

Wolves 25% (=20 scenarios out of 81)

Birmingham 47% (=38 scenarios out of 81)

Blackpool 53% (=43 scenarios out of 81)

Wigan 62% (=51 scenarios out of 81)

Thanks for the comment.

SeanThis is an interesting model and outcome. However surely it is overly simplistic, fundamentally flawed and tells us very little about the most likely outcomes on the football pitch this weekend.

On the positive side, this gives a very crude insight into who is most likely to go down, given the potential outcomes. But this model assumes that all outcomes are equally likely. In reality, they are far from it. To get a more realistic picture, there must be something built in that accounts for differences in quality.

For example, Blackpool travel to Man United, whereas Wigan are at Stoke. Stoke have a good home record, but where would you rather go needing a win to stay up? This could be done with some kind of points ranking based on league position. Although this would still not become totally accurate, we would end up with a much clearer picture by assigning some kind of likelihood of winning percentages based on the relative strength of the opposition etc? Because at the moment, this model isn’t going to predict anything…..

stevehaakeSean,

Yes, this is another way of doing it, by giving each score at each match a level of probability. As you point out, here we give every option equal probability and the chance of a 4-0 by Blackpool and a 4-0 win by Wolves would get very low probability compared to, say, a 1-0 win by Man U and 1-1 draw at Wolves.

However, the starting point is about how many basic scenarios are there which then allows a discussion. Ben above has done the most in-depth analysis so far and I would love to do a ranking system based upon league position. I’ve discussed this with others and there is yet to be a consensus on how to do the ranking and then account for the special nature of the final game of the season. Besides, I didn’t have time to do it!

However, what I’ve used the analysis for is to look at the combinations I think are the most likely based upon my knowledge of of the teams I’ve seen live this season (which is all of them). I have my prediction in a sealed envelope…

Thanks for the comments.

steveDo your sums again. Blackpool can lose and still stay up. So it ain’t ‘pretty obvious’.

(i.e. Birmingham also loses by quite a few goals making their GD worse than pools. And wigan lose too by either one less than Blackpool or more than one than Blackpool making either their GD worse or losing on goals scored).

stevehaakeSteve,

See new blog which addresses some of the comments about goal difference.

Thanks for your comment.

Survival Sunday in the Premier League: How to escape relegation. « The Centre for Sports Engineering Research[…] us and my earlier post on the number of scenarios appears to have stirred up feelings – see The Premier League: predicting who will go down (and the strange case of scenario no. 55). To recap, there are 5 teams separated by only one point who could all get relegated to the […]

What are the odds? | The Rovers Return[…] fine folk at Engineering Sport have come up with a mind-blowing bit of research that details all possible scenarios involving the […]

GJRThat is not correct if say Wigan and/or Blackpool lose narrowly say by a single goal and Birmigham lose heavily then Birmingham would br relegated. So Either Wian or Blackpool could lose and stay up on goal difference.

stevehaakeDear GJR,

Yes, you are quite right. I ignored goal difference to make it simple, but so many people have commented on this assumption I’ve had to do another article at:

https://engineeringsport.co.uk/2011/05/20/survival-sunday-in-the-premier-league-how-to-escape-relegation/

Have a read of that – I’m just worried that this will open up even more questions.

Ta for comment.

Should Premier League Clubs Bet On Their Own Demise? « Footy Finance[…] at risk after an extremely tight season has made it conceivable that a team could go down with 41 points. Relegation is much like Russian Roulette in that you are extremely excited for it not to happen […]

Premier League Relegation: the Wheel of Fortune « The Centre for Sports Engineering Research[…] our previous blogs we worked out that there were 81 different scenarios that could’ve taken place during the battle for the last two relegation places from the […]