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Finding a niche market in betting can often lead to value. This can be anything from specialist knowledge of handball betting to extensive experience in betting on Japanese baseball. Providing you know more than the bookmaker, there is money to be made.  &lt;Bet on soccer at Pinnacle&gt; They also make for interesting and potentially profitable bets, as they can be a little more reliable

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Poisson Distribution is a mathematical concept for translating mean averages into a probability for variable outcomes across a distribution. For example, if we know Manchester City average 1.7 goals per game, so by putting the Poisson Distribution formula tells us that this average equates to Manchester City&nbsp;scoring 0 goals 18.3% of the time, 1 goal 31% of the time, 2 goals 26.4% of the time and 3 goals 15% of the time.

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The Expected Value of a bet shows us how much we can expect to win (on average) per bet, and as such is the most valuable calculation a bettor can make when comparing bookmakers' odds. How can you calculate Expected Value in sports betting in order to predict your winnings? Read on to find out.

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Joseph is a betting analyst who manages the website www.Football-Data.co.uk, providing historical results, match statistics and betting odds data. He is also the author of Fixed Odds Sports Betting: Statistical Forecasting &amp; Risk Management (2003), How to Find a Black Cat in a Coal Cellar: The Truth about Sports Tipsters (2013) and Squares &amp; Sharps, Suckers &amp; Sharks: The Science, Psychology &amp; Philosophy of Gambling (2016).

Joseph Buchdahl

Much has been written in the days since Leicester City stunned the football world by winning the Premiership, with commentators, bloggers and modellers now attempting post hoc rationalisations to explain how Leicester’s against-all-odds climb to the top happened, why it happened and why it was always more likely to have happened then everyone previously expected.
In this respect, Leicester City’s Premiership title represents an example of what philosopher and statistician Nassim Nicholas Taleb calls a <a href="https://en.wikipedia.org/wiki/Black_swan_theory">‘Black Swan’</a>, a rare event that is hard to predict, comes as a surprise, but once observed, is explained in hindsight by a failure to account for relevant data that was available all the time. Reason is turned on its head, with outcomes being used to explain causes.&nbsp;
<h4>Was Leicester’s win predictable?</h4>
Just what were Leicester’s odds of winning the 2015/16 Premier League at the start of the season, and did the bookmakers underrate their chances? Of course, being a black swan event, attempting to extrapolate backwards to answer these questions runs the risk of <a href="https://www.pinnacle.com/en/betting-articles/betting-psychology/hindsight-bias-in-sports-betting">hindsight bias</a>. Nevertheless, let’s give it a go.
One possible way is to use the match betting odds market as a guide to how Leicester City were expected to perform. Taking Pinnacle’ Premiership match betting odds for the 2014/15 season and removing their margin, we can run a <a href="https://www.pinnacle.com/en/betting-articles/betting-strategy/how-to-analyse-your-betting-history-with-monte-carlo-simulation">Monte Carlo simulation</a> to determine the expected points total for all 20 teams, and from these their expected finishing positions too.
Whilst admittedly retrospective, the model is inherently Bayesian, reflecting changes of market opinion in the relative quality of teams, defined by the betting odds, as the season progressed. Some argue that a betting market is not necessarily accurate, since bettors suffer from behavioural biases, information cascades and group think.
<blockquote>A Monte Carlo simulation assessing the chances of Leicester City to become champions of the 2014/15 season showed a probability of 1 out 100,000.</blockquote>
But consistently such market inefficiencies are small, proving to be insufficient for the majority of punters to be profitable through anything other than chance. Moreover, given that a betting market is really a reflection of many hundreds or thousands of models, is there a good reason to doubt it being less wise than the mathematical analyses of a few so-called experts? The more diverse the set of opinions, the more wise the collective judgement is likely to be.
The table below shows the results of a 100,000-run Monte Carlo simulation. Leicester City were champions of the 2014/15 season in just one of those runs.&nbsp;
<table width="585" height="876">
<tbody>
<tr>
<td> 
Team
</td>
<td>
Expected points
</td>
<td>
Number of titles
</td>
<td>
Champions expected probability
</td>
<td>
Implied fair odds
</td>
</tr>
<tr>
<td>
Chelsea
</td>
<td>
81
</td>
<td>
46,682
</td>
<td>
46.682%
</td>
<td>
2.14
</td>
</tr>
<tr>
<td>
Man City
</td>
<td>
79
</td>
<td>
38,290
</td>
<td>
38.290%
</td>
<td>
2.61
</td>
</tr>
<tr>
<td>
Arsenal
</td>
<td>
72
</td>
<td>
9,462
</td>
<td>
9.462%
</td>
<td>
10.57
</td>
</tr>
<tr>
<td>
Man United
</td>
<td>
71
</td>
<td>
7,987
</td>
<td>
7.987%
</td>
<td>
12.52
</td>
</tr>
<tr>
<td>
Liverpool
</td>
<td>
67
</td>
<td>
3,293
</td>
<td>
3.293%
</td>
<td>
30.37
</td>
</tr>
<tr>
<td>
Southampton
</td>
<td>
60
</td>
<td>
338
</td>
<td>
0.338%
</td>
<td>
296
</td>
</tr>
<tr>
<td>
Tottenham
</td>
<td>
59
</td>
<td>
258
</td>
<td>
0.258%
</td>
<td>
388
</td>
</tr>
<tr>
<td>
Everton
</td>
<td>
56
</td>
<td>
91
</td>
<td>
0.091%
</td>
<td>
1,099
</td>
</tr>
<tr>
<td>
Swansea
</td>
<td>
48
</td>
<td>
3
</td>
<td>
0.003%
</td>
<td>
33,333
</td>
</tr>
<tr>
<td>
Stoke
</td>
<td>
46
</td>
<td>
1
</td>
<td>
0.001%
</td>
<td>
100,000
</td>
</tr>
<tr>
<td>
West Ham
</td>
<td>
45
</td>
<td>
1
</td>
<td>
0.001%
</td>
<td>
100,000
</td>
</tr>
<tr>
<td>
Newcastle
</td>
<td>
44
</td>
<td>
0
</td>
<td>
0.000%
</td>
<td>
N/A
</td>
</tr>
<tr>
<td>
West Brom
</td>
<td>
43
</td>
<td>
0
</td>
<td>
0.000%
</td>
<td>
N/A
</td>
</tr>
<tr>
<td>
Leicester
</td>
<td>
43
</td>
<td>
1
</td>
<td>
0.001%
</td>
<td>
100,000
</td>
</tr>
<tr>
<td>
Crystal Palace
</td>
<td>
41
</td>
<td>
0
</td>
<td>
0.000%
</td>
<td>
N/A
</td>
</tr>
<tr>
<td>
Hull
</td>
<td>
40
</td>
<td>
0
</td>
<td>
0.000%
</td>
<td>
N/A
</td>
</tr>
<tr>
<td>
Sunderland
</td>
<td>
39
</td>
<td>
0
</td>
<td>
0.000%
</td>
<td>
N/A
</td>
</tr>
<tr>
<td>
Aston Villa
</td>
<td>
39
</td>
<td>
0
</td>
<td>
0.000%
</td>
<td>
N/A
</td>
</tr>
<tr>
<td>
Burnley
</td>
<td>
36
</td>
<td>
0
</td>
<td>
0.000%
</td>
<td>
N/A
</td>
</tr>
<tr>
<td>
QPR
</td>
<td>
35
</td>
<td>
0
</td>
<td>
0.000%
</td>
<td>
N/A
</td>
</tr>
</tbody>
</table>
<h4>The past is the key to the future</h4>
Let’s assume that Leicester City would have been expected to perform more or less the same in 2015/16 as our model retrospectively expected them to perform in 2014/15. Despite winning 7 of their remaining 9 games last season to escape relegation, by August 2015 there was little other reason to expect any major change in expected performance.
Former manager Nigel Pearson, who had arguably engineered their Premiership survival, had left, Esteban Cambiasso, the club’s player of the year had departed, whilst new manager Claudio Ranieri had never won a league title, having recently been sacked by the Greek national team after losing to the Faroe Isles.
Most pundits and commentators were predicting relegation or skin-of-the-teeth survival. Even a <a href="http://talksport.com/football/premier-league-table-super-computer-predicts-final-bpl-table-201516-151007169511">super computer</a> predicted just 45 points and a 13th place finished for Leicester in October, when the team was already in 5th place.
Arguably then, in August 2015, 100,000/1 represented the fair odds for Leicester City winning the 2015/16 Premiership. Conceivably, bookmakers actually shortened their odds to 5000/1 on account of the Favourite–Longshot Bias.&nbsp;
<h4>Assesing Leicester’s performance for 2016/17 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</h4>
<blockquote>A 100,000-run Monte Carlo simulation based on this season’s match odds predicts Leicester to get an expected point total of 53 and an expected finishing position of 9th for the 2016/2017 season.</blockquote>
So what should Leicester City’s odds be for retaining the Premiership title in 2016/17? Another 100,000-run Monte Carlo simulation using this season’s match odds sees the team with an expected points total of 53 and an expected finishing position of 9th. Assuming, as before, that this season’s market opinion of the team will be broadly similar for the 2016/17 season, this translates into fair odds for becoming champions of roughly 625/1, considerably longer than the currently published odds ranging between 25/1 and 40/1.
Some have argued that it’s inappropriate to consider the whole 2015/16 season. Leicester obviously improved during the season, and since the betting market takes time (allegedly) to respond to a change in a team’s ability, wouldn’t it be better to consider the market’s view of Leicester in the second half of the season,&nbsp; once they were already leading the Premiership? Repeating the simulation now assuming the market had viewed Leicester in the first half as it had in the second sees them top just 1 in 111 times, an expected 8th place and 57 points, just 4 more than for the original simulation.&nbsp;
<h4>Skill or luck? – The final verdict</h4>
Finally, it’s worth observing that Leicester City’s expected points total for 2015/16 is just 10 and 12 more respectively than their expected (43) and actual (41) points total for 2014/15 season. This modest change in the betting market’s opinion about the team is a reflection of their genuine improvement in ability. 
By implication, the additional 27 points Leicester City accumulated has been achieved purely by chance. Given that the standard deviation in a team’s modelled expected points is typically about 7, their Premiership title truly represents a Black Swan.
Last season, the biggest difference between expected and actual points total was just 9 (Everton). If this modelling has any validity, it would seem we are still waiting for Leicester City’s performance to <a href="https://www.pinnacle.com/en/betting-articles/betting-strategy/regression-to-the-mean-in-sports-betting">regress to their mean</a>. They may be better than they were, but most of what has happened this season is arguably the result of good fortune.
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Considered 5,000/1 outsiders by bookmakers at the start of the 2015/16 season, the soccer world is still flabbergasted by Leicester’s City Premier League title. Could statistical analysis have predicted this rare event? Betting analyst, Joseph Buchdahl, takes a look at their performance to predict how well they are expected to do in the coming season. Here’s the verdict!