Correct Scores - binomial distribution

[The Hare]

The idea here is to see how well the binomial distribution performs as a predictor of the outcome of a football match. If we can estimate the expected number of goals that each side will score, and if the number of goals scored by each side follows the binomial distribution, (that's two big ifs!) then we can estimate the probability of every correct score combination. For example, typically we would expect the home team to score 1.5 goals. Using that figure, the binomial distribution tells us the probability of the home side scoring 0 goals is 0.22, 1 goal = 0.34, 2 goals 0.25 etc. Doing the same for the away side, who typically score 1.0 goals, allows us to caclulate all possible score combinations. I recognise that this must have been tried and no doubt failed by numerous people over the past, and I'd be interested to learn about others experiences. I've done some backtesting and the binomial distribution performs reasonably well, although there are some biases. To get this to work all I need is an estimate of the probability of a home and away goals in each match. I then look for value correct score bets - intitally I will taken anything where the value is +10% (prior to commission). All bets taken from Betfair. Will "dutch" my bets so that if any of my CS selections win, I win 100 units. Starting bank is 1000 units. This is very much a learning experience for me. Its my first go at a strategy, and dont have huge hopes of success, but hope to learn something on the way. Here goes: Stoke City v Coventry City CS Odds Stake 0-2 26 3.8 5 0-3 80 1.25 Inverness CT v Aberdeen CS Odds Stake 0-0 10.5 9.52 0-1 9.6 10.42 1-0 8.6 11.63 2-0 15 6.67

[dr.karl]

Re: Correct Scores - binomial distribution It's been a long time since I've used the binomial distribution but I fail to see how it works in this case. How have you calculated the chance of there being one goal scored by the home team if the team is expected to score 1.5 goals? The way I thought a binomial distribution worked was with combinations of "yes/no" factors occuring, so for example a coin toss, its either head or a tail. If you wanted to know the probabilty of tossing 13456 heads out 20000 throws this is the distribution to use. Could you explain the maths behind it please because it'll bug me for days if I can't figure it out! Oh and gl with the system :ok.

[slapdash]

Re: Correct Scores - binomial distribution I think he means Poisson distribution?

[The Hare]

Re: Correct Scores - binomial distribution I'll try and clarify my thinking, and hope this does not lead to more confusion. By way of example, suppose for simplicity that there are 100 mins in a football game, and that the expected number of goals the home side scores is 1. One way of thinking about this is to consider every minute of a game a unique event, where the home side has a 1/100 chance of scoring a goal, and a 99/100 chance that they wont. We therefore have a binary event. We do exactly the same with the away side. In other words the number of goals that the home team scores and the number the away team scores are assumed to follow two independent binomial distributions. For example, if both teams have an expected number of goals = 1, the probability of a 0-0 draw is: (99/100 )^100 * (99/100)^100 = 0.134 To give another example, the probability of a 2-0 home win would be: (99/100)^98 * (1/100)^2 * 100!/98!2! * (99/100)^100 = 0.068 As it happens I have used two independent binomial distibutions to calculate the expected probabilities of each correct score. As Slapdash rightly points out, I could have used the Poission Distribution. However, when the probability of an event is very low (which it is here) the poission and binomial distributions converge, and give almost identical probabilities. It is easy to see the flaw in this system. The calculations are based on the assumption that the probability of one side scoring a goal is independent of the number of goals scored by the other side. It is easy to think of examples where this does not hold. For example, if one side is winning 1-0, and the other is chasing the game, the match may become more open with an increased likelihood that both sides will score. However, equally, the side in the lead may prefer to sit on the lead, and close the game down, so that the probability of a goal is reduced. The idea behind this thread is really to see how well the binomial/poission distribution performs. My backtesting, admittedly limited, has revealed that most correct score odds are very close to those predicted by the binomial/poission distribution. However, occasionally there are some very significant outliers, and the aim of this thread is to see whether these might offer genuine value, or whether there are very sound reasons why the market expects these correct scores to deviate from the binomial/poission distribution. Unfortunately, many of these are very long shots, and I doubt whether I will have the time or patience to sustain this thread in order to know whether this is likely to be an effective strategy.

[The Hare]

Re: Correct Scores - binomial distribution First two bets lose. Not expecting a high strike rate , but it would have been nice to strart with a winner. One bet for tonight: Southend v Manchester United back unquoted score (one team 4 or more goals) @6.8 (stake =14.71). Not confident but system suggests 20% value. Incidently, if I ever manage a winning bet commission will be deducted at 5%.

[pinhofap]

Re: Correct Scores - binomial distribution Good Luck, it's always nice to see someone following a mathematical and disciplined approach to betting, not subjective factors. I hope you do well with this. Pinho

[The Hare]

Re: Correct Scores - binomial distribution Thanks for the good wishes Pinofap. Last night was another loss, so still waiting for first success - will provide an update when thanhen that happens. One silver lining was that most of the bets that the system identifies as value have come in significantly. This may provide an arb opportunity or there again I may just be clutching at straws! Tonights selections are: Birmingham City v Liverpool Back 2-0@38 (stake 2.63) Back 3-0@140 (stake 0.71) Hibs v Hearts Back 2-0@19 (stake 5.26) Back 3-0@42 (stake 2.38) Back 0-2@18 (stake 5.56)

[The Hare]

Re: Correct Scores - binomial distribution In an attempt to secure first winning bet, will try German football tonight. Werder Bremen v Dortmund. Back: 0-1@27 Stake = 3.70 0-2@65 Stake = 1.54 1-2@25 Stake = 4.00 1-3@80 Stake = 1.25 2-3@95 Stake = 1.05 Obviously these are a series of long shot bets. This is equivalent to a bet at odds of 8.66. Total stakes are 11.54, and any winning CS returns a profit of 100 units.

[The Hare]

Re: Correct Scores - binomial distribution Well, my foray into German football provides my first win - this was starting to get embarassing! (Werder Bremen 1 Dortmund 3). Here, as promised is the profit and loss update for the thread: Starting bank 1000 Current bank 1008.94 Total staked 86.13 Number of bets 6 Winning bets 1 Losing bets 5 Strike rate 17% Yield 10% Profit/loss 8.94 Average odds 6.97

[The Hare]

Re: Correct Scores - binomial distribution First selection for the w/e - Man City v Newcastle. The model predicts that the following CSs are value: 0-3@95 (Stake 1.05) [email protected] (5.41) 1-3@60 (1.67) 3-0@28 (3.57) Unquoted@19 (5.26)

[The Hare]

Re: Correct Scores - binomial distribution Three games for Sunday: Reading v Spurs 0-0@10 Stake = 10 0-1@8 = 12.5 0-2@13 = 7.69 0-3@34 = 2.94 3-0@75 =1.33 Arsenal v Liverpool 0-2@30 STake =3.33 0-3@100 =1 1-3@65 =1.54 Unquoted@21 =4.76 St Mirren v Celtic 0-0@18 Stake = 5.56 [email protected] =13.51 [email protected] =12.82 [email protected] =8.7 Feeling a litttle more confident about these than yesterdays feeble effort. Tme will tell!

[The Hare]

Re: Correct Scores - binomial distribution A very poor w/e for the system with no winners out of the four games selected. One bet for tonight in the SPL: Falkirk v Hearts Back:

0-2 @12.5 (stake = 8.00)

0-3 @24 (stake = 4.17)

2-0 @27 (stake = 3.70)

[The Hare]

Re: Correct Scores - binomial distribution I do hope that nobody is seriously following these selections as the current yield is -56%. However, as the average odds is over 5.0 it would just take one bet to take this significantly into profit. Way too early to make any judgment about the success of this strategy - but a win would be nice. Tonights selection is: Wales v Liechtenstein Back: 0-1 @ 60 (stake = 1.67) 1-0 @9 (stake = 11.11) 2-0 @ 7.2 (stake = 13.89) 3-0 @ 7.2 (stake = 13.89) Incidentally, according to the sytem, the best value bet tonight is under 2.5 goals - just watch those goals roll in!

[The Hare]

Re: Correct Scores - binomial distribution AAAgh - last minute goal by Wales last night denied the system the winner that would have taken it into profit. Tonights selections are: Holland v Eire back 0-2@ 23 (stake = 4.35) 3-0 @ 34 (stake = 2.94) Eire v San Marino back unquoted @ 1.44 (stake = 69.44) (one side score 4 or more goals) Note that stakes are, as allows, proportionate to the odds, so that any correct score that wins returns 100 units (although this is not 100 units of profit)

[The Hare]

Re: Correct Scores - binomial distribution Eire provided second winner but at skinny odds so system remain in loss (will update after this e/e selections). Two to kick off: Manchester C v Fulham Back: 0-2@25 (stake = 4.00) 0-3@85 (stake = 1.18) 3-0 @29 (stake = 3.45) unquoted@20 (stake = 5.00) Middlesbrough v Liverpool Back: [email protected] (stake 13.16) 0-3@27 (stake 3.70) Not that confident, but fingers crossed

[The Hare]

Re: Correct Scores - binomial distribution Three more - all backs: Ipswich v Norwich 0-1@12 (stake = 12) 2-0@16 (6.25) 3-0@36 (2.78) Wigan v Villa [email protected] (10.42) 0-2@20 (5.00) 0-3@60 (1.67) Hearts v Rangers

[The Hare]

Re: Correct Scores - binomial distribution Hearts v Rangers - back: 0-3@38 (stake = 2.63) 3-0@ 50 (stake = 2.0)_

[The Hare]

Re: Correct Scores - binomial distribution Well, all selections over the w/e lost. The Man C result was particulary galling, as, at half-time it looked a racing certainty. The only consolation for the poor start to this system is that nobody appears to be looking in!

Starting bank	1000
Current bank	802.04
Total staked	391.49
Number of bets	19
Winning bets	2
Losing bets	17
Strike rate	11%
Yield	-51%
Profit/loss	-197.96
Average odds	4.85

It is a long odds system (average = 4.85), so it is easy to make such a dismal start. I may start another thread looking at the best value lays, as this should reduce the variability of the yield. Hoping Champions League will change my luck. Tonights selections are (all backs): Celtic v Man U 0-3 @ 24 (stake = 4.17) 2-0 @ 28 (3.57) 3-0 @ 110 (0.91) 3-1 @ 60 (1.67) UQ @ 16 (6.25) Arsenal v Hamburg 1-2 @ 110 (stake = 0.91) 1-3 @ 360 (stake = 0.28) 2-2 @ 100 (stake = 1.00) 2-3 @ 500 (stake = 0.2) 3-3 @ 400 (stake = 0.25) CSKA v Porto 2-0 @ 13 (stake =7.69) 3-0 @ 34 (2.94) 3-1 @ 30 (3.33) 3-2 @ 75 (1.33) UQ @ 25 (4.00) Will be very surprised if the Arsenal game in particularl comes up trumps, as it is effectively amounts to staking 4 units at odds of 25.

[Sgt.Sunshine]

Re: Correct Scores - binomial distribution Hare - it has been shown many times in the stats literature that the Poisson produces a good fit to footy scores, but under- or over-estimates some scores consistently, from memory 0-0, 1-1 and a couple of others. I also seem to remember that using a double binomial (?) overcomes this problem.

[The Hare]

Re: Correct Scores - binomial distribution Thanks for the feedback and info Sgt. Sunshine - much appreciated. I'm using a double binomial, at least I think I am in that I'm using a separate binomial for the home and away teams, so it will be interesting to see if it overcomes some of the problems you say exist in the poisson distribution. Would like to learn more about this. The theory behind the thread is that, although not perfect, the binomial should be a reasonable approxomation to the final score, hence, I only take odds which appear to be significantly out of line. Will do some detailed research when I have more data. Two patterns do appear to be emerging, (1) it is good at predicting over/unders, but (2) it underestimates the probability of draws, particularly 1-1. The frustrating thing here is that many of the CS's seen as value are very long-odd shots (helped by the low overound on Betfair) so it takes ages to come to any sort of conclusion as to whether this is likely to work or not.

[The Hare]

Re: Correct Scores - binomial distribution To celebrate my first piece of feeback in a fortnight here are 5 selections from tonight (all backs, and all Champions League). The one interesting game for the system is Werder Bremen v Chelsea, where the binomial suggests that there is excellent value in an away win. We shall see! 1. Spartak Moscow v Bayern Munich 2-0 @ 25 (stake = 4.00) 3-0 @ 95 (1.05) 2. Liverpool v PSV 0-3 @ 190 (stake = 0.53) 1-2 @ 27 (3.70) 3-0 @ 24 (4.17) UQ @ 18.5 (5.41) 3. Levski Sofia v Barcelona 1-0 @ 60 (1.67) 1-1 @ 19 (5.26) 1-2 @ 13 (7.69) 2-0 @ 220 (0.45) 2-1 @ 70 (1.43) 2-2 @ 60 (1.67) 2-3 @ 70 (1.43) 3-0 @690 (0.14) 3-1 @ 610 (0.16) 3-2 @ 310 (0.32) 3-3 @ 27 (0.37) 4. Inter Milan v Sporting Lisbon 0-2 @ 60 (1.67) 0-3 @460 (0.22) 1-2 @ 46 (2.17) 1-3 @200 (0.5) 2-2 @44 (2.27) 2-3 @180 (0.56) 3-3 @220 (0.45) UQ @ 9.2 (10.87) 5. Werder Bremen v Chelsea 0-2 @ 19 (5.26) 0-3 @ 50 (2.00) 1-2 @ 14 (7.14) 1-3 @ 38 (2.63) 2-0 @ 17 (5.88) 2-3 @ 50 (2.00) 3-0 @ 44 (2.29) UQ @ 16 (6.25)

[The Hare]

Re: Correct Scores - binomial distribution Sorry for the absence of updates - safe to say this thread is very seriously unprofitable at the moment with a yield of -50%. Hardly promising, but I'm not so easily deterred. Two more allegedly value bets, including 0-1 to the Arsenal. Satrdays correct score backs are: Charlton v Everton 0-2 @16 (stake = 6.25) 0-3 @ 46 (stake = 2.17) Bolton v Arsenal 0-1 @ 8.4 (stake = 11.9) 2-0 @ 24 (stake = 4.17) A win would be nice

[The Hare]

Re: Correct Scores - binomial distribution 3 more for Sunday: Newcastle v Portsmouth (backs) 0-0 @ 9.2 (10.87) 0-1 @ 9 (11.11) 0-2 @ 21 (4.76) 0-3 @ 70 (1.43) Hibs v Celtic 0-2 @ 13.5 (7.41) 1-0 @ 22 (4.55) 2-0 @ 44 (2.57) 2-1 @ 21 (4.76) 3-1 @ 60 (1.67) 3-2 @ 53 (1.82) Man U v Chelsea 0-2 @ 19.5 (5.13) 0-3 @ 60 (1.67) 3-0 @ 40 (2.50) UQ @ 30 (3.33) Incidently, the odds of 30 for unquoted (one or more sides scores 4 or more goals is the highest by some distance since I have been tracking matches - no doubt for a good reason

[Cloud]

Re: Correct Scores - binomial distribution I'd like to add my insight into the Poisson Distribution idea. I ran a Poisson distribution based upon average goals for home and away teams - and then compared the results to the actual results of the game (and I did use seperate poissons for home and away). And yes - the 1-1 was a lot more popular than it should be (I put this to team mentality btw) - around 9%. So i had an idea. I thought I would compare the actual frequency with the "calculated" frequency for each score. From this I got figures for how much more likely/less likely a score would be compared to Poisson - and then altered the stats accordingly. For example - My stats told me that 1-1 was 9% more likely than it should be. So, if the poisson told me that 1-1 was a 13% chance - I would multiply this by 1.09, so it would become a 14.17% chance. Some other scores were therefore less likely - and I would alter the percentages accordingly. Now I did this for a small sample, and it didn't really work - but my sample size was far too small. Not having the time to run a longer test, I am none the wiser as to whether this works - but on the face of it, I think that this seriously needs looking at. If anyone would like to run with this - feel free. And if it works - let me know. Just to add to this - I used the spread firms to get figures for predicted goals for each team - using a combination of total goals, and team supremacy.

[muppet77]

Re: Correct Scores - binomial distribution i did exactly this for 4713 games - think it was epl. i did it for the goals scored, rather than the scores. multiply goal% to get score%s. here's what i got:

	score	adjust	poisson	observed
home	0	1.06	1039.52	1105
goals	1	0.99	1571.30	1548
	2	0.96	1187.56	1139
	3	0.94	598.36	560
	4	1.06	226.11	240
	5	1.24	68.36	85
	6	1.28	17.22	22
	7	3.23	3.72	12
	8	1.42	0.70	1
	9	8.47	0.12	1
	score	adjust	poisson	observed
away	0	1.05	1559.30	1637
goals	1	0.98	1724.72	1685
	2	0.90	953.85	858
	3	1.06	351.68	373
	4	1.26	97.25	123
	5	1.12	21.51	24
	6	2.77	3.97	11
	7	1.60	0.63	1
	8	11.54	0.09	1
	9	0.00	0.01	0
				4713

[Cloud]

Re: Correct Scores - binomial distribution Great minds think alike!!! Any success???

[muppet77]

Re: Correct Scores - binomial distribution in a word - no. poisson is superb for prediction. however, the key is getting the correct value for the goal 'average' that you input into poisson. rubbish in, rubbish out.

[Cloud]

Re: Correct Scores - binomial distribution Looking at your figures - it seems that the adjustment for the 1-1 draw would make that happening less likely, rather than more likely. If you work out the distribution based on overall scorelines - I think you get a more accurate picture. And yeah - I fully agree - crap in - crap out. As I said I used the spread firms (an average of 3) - so I was hoping there would be some accuracy there. As I said - it still hasn't been tested.

[The Hare]

Re: Correct Scores - binomial distribution First things first, here is an update of the disastrous picture to date:

Starting bank	1000
Current bank	580.48
Total staked	613.05
Number of bets	32
Winning bets	2
Losing bets	30
Strike rate	6%
Yield	-68%
Profit/loss	-419.52
Average odds	5.22

Many thanks for the terrific feedback, and stats. There does seem to be a consensus that the poission is prettty good at predicting, other than 1-1, with the very important provisio that it relies on accurate forecasts of the number of goals scored (garnage in, garbage out). I've been working with separate estimates of the number of goals scored by the home and away team. I too use the prices of the spread firms as a guide, but I do use other information sources to come to what i believe is a more accurate figure. Its a really small sample here, and luck has certainly not gone my way. On closer inspection it is quite easy to see why. The average number of goals in my games chosen so far has been really low at only 2.25. The spread firms figures for the same matches was 2.66! Interestingly, I correct for this "bias", then my strike rate rises from 6% to 44%, and my yield to a more healthy +5% (after commission) - which is my target figure. So, it does seem as though the problem so far is in obtaining a reliable estimate for the number of goals scored in a match, rather than the Poisson distribution itself (with the exception of 1-1 draws). The feedback has encouraged me to perservere with this, although I think I may revise what type of bets I go for. I still feel that correct scores offers a larger number of value bets than other combinations - for example, the system often predicts "no bet" for overs/unders.

[Sgt.Sunshine]

Re: Correct Scores - binomial distribution I got the terminology wrong - the paper I read used a bivariate Poisson. This is different to what you are doing I think, which is using two separate distributions for home and away, and multiplying the output to give a probability. The bivariate Poisson allows for correlation between the two teams' scores, rather than treating them separately. I have one paper I can pm to you if you want, although it's not the one I was originally thinking of where they found the discrepancies for 1-1 etc. The problem with using Poisson is that it's easy and it's (presumably) what the odds compilers use, so the chance of getting rich using it to predict correct scores are pretty slim, even allowing for good odds on Betfair.