How to calculate my edge here?

[dmolition]

Im confused about a prop for first score on a soccer match
the prop is this:

under 27 minutes +110
over 27 minutes -111
No score +700

One team averages a first score at 38.25 minutes and 38.33 minutes the second team
Can my edge be calculated by poisson distribution on this instance or it doesnt apply?
How would you give the total given the averages since 27 seems so low compare to both team averages i may be missing something.

THanks.

[sharpcat]

whats your estimated probability of the 2 teams scoring in less than 27 minutes?
whats your estimated probability of the 2 teams scoring in more than 27 minutes?
Whats your estimated probability of no score occurring?

Knowing the average time range for when the first goal occurs does not take into consideration the probability of each sides occurrence nor does it consider the probability of no score at all.

[Peep]

I would think you would want to factor in the "gives up a first score" for both sides as well.

[dmolition]

I would think you would want to factor in the "gives up a first score" for both sides as well.

Well i thought about this but as it turns out no TEAM gives up a score before they "first score" so i skipped that from the analysis, im still confused by the probabilities though.

Pinnacle has the score 0-0 for regular time at +821 which is 10.12% no vig that leaves 89.88% being under or over 27 minutes. Im not sure how to go from there do i need to count all the scores before 27 minute and all scores after 27 minutes (no score lands exactly at 27 in the stats) and see the percentages and from that my 100% will equal the remaining 89.88%?? Do i need to factor in all scores in all matches of the league also to be more accurate or just these two teams?

Thanks for the response.

[byronbb]

The averages are consistent with O/U 2.5 goal line.

Can't we just expect 0.7 goals per 27 minutes then poisson it to no vig of 100?

(how well does Poisson do with numbers less than 1?)


edit if you look at 1st half 2nd half numbers they weight goals towards the 2nd half by about 60% (like a 2.5 goal line will have 1st half +/-1 and 2nd half +/-1.5.

edit ---

If we use Pinny's +821 for no goal leg instead of the +700, that implies fair pricing at +132 for the under and +111 for the over.

[Ganchrow]

There are certainly some serious problems with modeling the bet you've outlined using a pure homogeneous Poisson process. Off the top of my head:

• The information given completely neglects the relative strengths of opposing offenses/defenses.
• One might expect there to be some negative correlation between opposing teams' offensive (and defensive) first scores (especially earlier in the match).
• A Poisson rate wouldn't take into account either end-of-match effects or start-of-match effects (to the extent the latter exists, or which I'm admittedly not certain).
• There are likely other memory effects at play (which by definition don't conform to a Poisson process). For example the larger the point differential the more likely it might be for the winning/losing team to focus on offense/defense as the match progresses. Now while these effects wouldn't appear to impact the time of first, what they do illustrate are the general inappropriateness of the Poisson
• Due to injury time, it might not be strictly appropriate to compare time spans between match.

And I'm sure all of us here in the Think Tank could come up with numerous other objections to using a Poisson process in such a fashion

Now with that caveat well behind us, let's see what we can come up with using just the Poisson along with the following information (please correct if I misstated any of this):

First score in regulation+injury time under 27 minutes +110
First score in regulation+injury time over 27 minutes -111
No score in regulation 10.12% (taken as a given)

Note that because we're assuming scoring to represent an homogeneous Poisson process it then follows (within our pretend world) that scores for either team occur (on average) at equally spaced intervals within the game. Now again ... we do know this to be false in reality but for the sake of this first order approximation we're assuming otherwise.

You'll also note that we're ignoring the historical first score information. The reason is simply that as long we're assuming a Poisson process (and sticking to it), there'd be no way to reconcile those information nuggets with Pinnacle's 0-0 score no-vig odds.

Recall from basic probability that the exponential distribution models stopping time between Poisson-distributed events. So using a resolution of seconds and based upon what we'll assume to be 5,820 seconds of relevant game time (that's 97 minutes of regulation+injury time):

F(5820 + ε; λ) = 1 - 10.12%

Where F(x, λ) corresponds to the CDF of the exponential distribution:

F(x, λ) = 1 - e^-λx)

and ε corresponds to an arbitrarily small number > 0.

Solving for the rate parameter λ:

F(5820; λ) = 1 - e^{-λ * 5820} = 1 - 10.12%
λ ≈ 0.03935836% goals per second

It's now a simple matter to plug the solution λ value of 0.03935836% back into the exponential distribution for t = 27 minutes - 1 second:

F(27*60 - 1, λ) = 1 - e^{-0.03935836% * (27*60 - 1)} ≈ 47.1235%

And now for 27 minutes ≤ t ≤ 97 minutes:

Pr(27 minutes + 1 second ≤ t ≤ 97 minutes) ≈ 1 - 10.12% - F(27*60 + 1, λ) = 1 - 10.12% - 1 + e^{-0.03935836% * (27*60)} ≈ 42.7357%

Now because we're excluding t = exactly 27 minutes (which ostensibly would be a push on all 3 bets) we the divide through each of our absolute three probabilities by the sum of the absolute probabilities to yield a final conditional probability for each of our 3 relevant outcomes:

P(t < 27 minutes) ≈ 47.1235% / 99.9792% ≈ 47.1333% → +112 fair value
P(27 minutes ≤ t ≤ 97 minutes) ≈ 42.7357% / 99.9792% ≈ 42.7446% → +134 fair value
P(> 97 minutes) ≈ 10.1200% / 99.9792% ≈ 10.1221% → +888 fair value

So based on this admittedly extremely simple analysis that uses nothing other than the Pinnacle no-vig 0-0 line as a model parameter, we'd conclude that the given market provides no profit opportunity on any of the 3 wagers. It's interesting to note, however, that the under appears priced at near zero-vig, while the over appears at almost 18.75% vig.

Absent any a priori rationale for such a pricing discrepancy, and depending upon the book and limits at which this prop was found this should probably give us some pause as to the general validity of our model assumptions.

Anyway, this simple analysis is no more than a quick and dirty back-of-the-envelope calculation. It ignores numerous factors undoubtedly at play. Personally, I'm not all that concerned by its neglect of historical first score time averages in favor of the Pinnacle 0-0 line in and of of itself. The rationale for that would be that such historical point data completely neglects strength of opposition, which is information we'd expected top be encapsulated within the Pinnacle line.

What is of most concern would be the underlying assumption of soccer goals following a Poisson process. Not only should the theoretical objections to such a characterization be clear and apparent, but there also happen to be a wealth of economic papers on the subject that demonstrate that soccer scores historically do NOT follow a homogeneous Poisson process.

Nevertheless, I feel that the above represents a decent first-order approximation, making use of nothing more than the information given. Someone interested in props of this sort might be well served b first utilizing this miserly model to provide baseline historical estimates and then examining the biases to help determine the relevance of additional data as well as a more suitable model specification. (The simplest upgrade to the model itself would probably involve some sort of non-homogeneous Poisson process where the rate parameter varied with time in a manner consistent with both experimental observations and cogent underlying theory.)

Anyway, I do hope that this provides an easily digestible hors d'oeuvre for thought. I'd expect that with just a little added thought and legwork on top of the above Poisson-based analysis a prop bet such as this should be readily beatable at a good number of traditional US-facing books.

[lasker]

F(5820 + ε; λ) = 1 - 10.12%

Where F(x, λ) corresponds to the CDF of the exponential distribution:

F(x, λ) = 1 - e^-λx)

and ε corresponds to an arbitrarily small number > 0.

Solving for the rate parameter λ:

F(5820; λ) = 1 - e^{-λ * 5820} = 1 - 10.12%
λ ≈ 0.03935836% goals per second

It's now a simple matter to plug the solution λ value of 0.03935836% back into the exponential distribution for t = 27 minutes - 1 second:

F(27*60 - 1, λ) = 1 - e^{-0.03935836% * (27*60 - 1)} ≈ 47.1235%

And now for 27 minutes ≤ t ≤ 97 minutes:

Pr(27 minutes + 1 second ≤ t ≤ 97 minutes) ≈ 1 - 10.12% - F(27*60 + 1, λ) = 1 - 10.12% - 1 + e^{-0.03935836% * (27*60)} ≈ 42.7357%

Now because we're excluding t = exactly 27 minutes (which ostensibly would be a push on all 3 bets) we the divide through each of our absolute three probabilities by the sum of the absolute probabilities to yield a final conditional probability for each of our 3 relevant outcomes:

P(t < 27 minutes) ≈ 47.1235% / 99.9792% ≈ 47.1333% → +112 fair value
P(27 minutes ≤ t ≤ 97 minutes) ≈ 42.7357% / 99.9792% ≈ 42.7446% → +134 fair value
P(> 97 minutes) ≈ 10.1200% / 99.9792% ≈ 10.1221% → +888 fair value

So based on this admittedly extremely simple analysis...

There's no hope for me

[Ganchrow]

There's no hope for me

I didn't mean to suggested that I explained myself in simple terms, but rather that my analysis was simplistic.

A while back Justin7 posted a video on SBR.tv about the Poisson distribution and its use in prop bet analysis. Definitely worth a viewing.

[trixtrix]

ganch, good to see you back to posting buddy, for awhile there i was concerned that you were having some health issues or something that prevents you from posting

are you back in the U.S or still in cr?

[Wrecktangle]

I'm certainly not a soccer punter, and I might not even understand the question, but short of getting a play-by-play history of soccer games, I don't see how the distribution of goals over time in a soccer match differs from a flat distribution (any minute as equally likely as the next) scaled by a team strength factor as Ganch says. Further, I would think there is some lead up time in the first few minutes where goals are less likely similar to NFL or college football scoring. The distribution may fall off due to teams getting tired later in the game, but poisson seems to not apply to dist. of goals over time during a match.

As I mentioned in earlier posts, sports are full of special distributions that only fit that sport. In order to be confident, in this case you must obtain a large db to measure. Due to the infrequence of scoring and differing team strength it may have to be as large as a minimum of 1000 games.

[lasker]

I didn't mean to suggested that I explained myself in simple terms, but rather that my analysis was simplistic.

A while back Justin7 posted a video on SBR.tv about the Poisson distribution and its use in prop bet analysis. Definitely worth a viewing.

Thanks very much, I'll take a look.

[dmolition]

Great reply ITT by Ganchrow and others, this is for the WC so there arent a lot of games i made a distribution of all scores and yes 72% of the scores are past the 27 minute, due to other factors i would think that may include fatigue and change of tactics, etc, but this distribution is accounting all scores and not just "first scores" so i dont know if its relevant. Also i realized poisson distribution analysis is not clear cut for the problem.
I checked the times a 0-0 score ended in the 90 minutes and is almost consistent with Pinnys percentage,
I would think is better to use the number of games that ended 0-0 than the pinny no vig line, right?

Thanks for the respones i have to read a lot know to understand it well.

[Ganchrow]

I don't see how the distribution of goals over time in a soccer match differs from a flat distribution (any minute as equally likely as the next)

Actually, that the probabilities of a score (or more generally, any given number of scores) during any time intervals of equal length were equivalent would be a necessary condition for an homogeneous Poisson process.

That said, modeling scoring probabilities as you've suggested would actually yield results essentially identical to those of the Poisson process outlined above.

Using the Pinnacle 0-0 probability of 10.12% and assuming a uniform score probability distribution over time, let's say we divide up the relevant game time into k equally sized non-overlapping scoring intervals.

Because we're assuming uniform, the probabilities of a score during all such intervals are necessarily identical. Call that probability p_k.

Further, the only way for there to be no score after all k intervals (p=10.12%) would be were there no score during any of these intervals.

This implies

p = p_k^k
p_k = p^1/k
p_k = 10.12%^1/k

Hence over the course of n time intervals what we have is simply a sequence of n Bernoulli trials with success probability (success being no score during the given interval) = p_k.

Extrapolating, the probability of at least one score over the first 27-ε out of 97 minutes (≈ 27*(k-1)/97 periods) would be:

P(t < 27 minutes) = 1-10.12%^{27(k-1)/(97k)} = 47.139%

Which is practically identical to the probability yielded by the Poisson process and asymptotically approaches equivalence as the time resolution approaches 0 (the difference lies with the conditioning).

Of course none of this should be interpreted as lending any credence to either of the distribution characterizations themselves, but should only serve to illustrate that the two in practice produce identical results.

[Ganchrow]

ganch, good to see you back to posting buddy, for awhile there i was concerned that you were having some health issues or something that prevents you from posting

are you back in the U.S or still in cr?

Rumors of my death were greatly exaggerated.

And it's Costa Rica all the way.

[SolidDala]

In WC there is a few other variables to take into notice:

In the first round; amount of goals scored under 27 min is far less then the final round of group stage. This is mainly because teams play not to lose in 1st round, and in the final round these teams ( dogs ) have to win, meaning they most attack, opening up their defense a lot more.

[dmolition]

Ganchrow i have a question,

seeing that total goals in soccer are not suitable for poisson distribution analysis would the same apply to total corners in match?

Iif it does apply how would you do the analysis in a World CUp being that tactics and teams change from group play to knockout stage, for example in knockout stage the average or corners per game is 9.44 and the median is 9, lots of sites have a prop total at 9.5, but im guessing is not as simple as that, i may have to take account of both teams and defenses, corners allowed vrs executed, etc.

thanks for the response.

[Ganchrow]

seeing that total goals in soccer are not suitable for poisson distribution analysis would the same apply to total corners in match?

I'd expect to see many of the same problems analyzing corners via the Poisson.

[Justin7]

Poisson is not very good for corners. How often do you see a corner attempt, immediately followed by another one? If the occurrence of one increases the likelihood of another for a short period of time, pure Poisson does not work.

[dmolition]

thanks for the responses this thread has got me reading statistic for dummies I and II

so poisson is more aplicable to Basketball (rebounds, asts,etc) NFL, NCAAFL (passes, interceptions, fieldgoals)
than soccer stats.
if soccer stats dont work with poisson i need to take a look at rates, binomial distribs, etc?

Thanks again, also any book recommendation to refresh my statistics course would be great.

[aggieshawn]

boy this is good.

[Wrecktangle]

I guess it is the engineer in me. I always prefer to work from the data before I assume a distribution. In some cases it is obvious, but I always think you should check to be sure. This always involves building some sort of db, usually a massive one.

In this case, short of having and analyzing the data, perhaps a "long" poisson with it's tail chopped will work as it might capture the rise at the beginning, but those typically don't rise fast enough. I would capture a lot of data before I threw anything down on it. That said, as a quick and dirty, "let's do something to *convince* myself I'm making a decent bet" sort of thing, why not?

I always part quickly from the classical statisticians as they always seem to want to assume a distribution going in, on the other hand I want to the data to convince me. Reading Tukey got me there.