Solved – Probabilities of betting odds not adding up to 1

conditional probabilitylogisticprobability

I am currently studying logistic regression. So the Probability(p(x)) of assigning an outcome is calculated as :

$\frac{p}{1-p} = e^{\beta_0 +\beta_1X} $

Further, I read that the fraction on the left is also called odds, frequently used in betting references.

So I went to a betting website, here were the odds given by them for a football match.
Chelsea vs West Brom ( Win Draw Win : 1.5, 4, 7.5)

Now I understand that they would have used a program to come up the odds.

But I figured that using the odds, I should be able to calculate the probabilities, that their software is assigning to each of the outcome.

When doing the calculations : I got 0.4 prob of Chelseas win, 0.2 of Draw and 0.11 chance of West Brom Win.

Why is this not adding up to 1?

Best Answer

Decimal odds can be turned into their implied probabilities via the formula: $P=1/odds$. If you do this and sum you get 1.05 so I am guessing there is a small rounding errors in the decimal odds reported.

In response to the comment below, yes I am sure since these are,decimal odds . What is missing in this formula is the overhead that a "bookie" often uses to make the odds lower.

This formula is: $odds = 1/(P + \sigma)$, If you assume that the $\sigma$ are the same you can calculate this bias, and the probability will sum to 1.

Also see this post: How to convert sport odds into percentage? Asking essentially the same question.

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Solved – How do betting houses determine betting odds for sports

How odds are set is a really interesting subject that I have done some research into, and in a similar way sports analytics.

The first paper I would refer to covers the NFL specifically "Why are Gambling Markets organised so differently from Financial Markets", Steven.D.Levitt (The Economic Journal 2004). This illustrates that the odds on the NFL are rarely set to generate 50/50 action because the bookmaker can exploit "square" action by skewing odds against their traditional bias (i.e. the point made above about the Ohio State Buckeyes - if the bookmaker is aware that they are going to take a larger % of the bets, they can either adjust the odds or the spread so the better has to pay a premium to bet the Buckeyes - e.g. the -7.5 or more than one touchdown instead of -6.5 - especially if the true rating for the game was around -5 or -6). It also makes the point that bookmakers/sportsbooks rarely make the odds themselves they are usually paying influential odds makers who set the line for a lot of events. The bookmaker will then rarely adjust these odds greatly as they will effectively handicap the market against other bookmakers and sportsbooks (generating a profitable opportunity for "sharp" action).

In the case of the game quoted by the OP, the prices quoted by Bet 365 is consistent with the over-round % that they have run on most football games this season of between 105-107% (I have an interest in this - their over-round% on the English Premier League is typically 5-6%). That 5-7% margin will look after them in the long run as it increasingly means unsophisticated gamblers have to be more right than average in the long run to make a a sustained profit. How the actual odds are generated is another matter in the case of Bet365 a lot of their competitors use the Bet Genius group for Odds Data (e.g. Sportingbet, Paddy Power, Sky Bet). They will probably then make small adjustments to this based on their typical clients betting preferences (e.g. what type of action they take and biases).

For a lot of sports the Cantor Fitzgerald group have created the Midas Algorithm to set up odds in the same way they would deal on Wall Street and they are getting an increasing presence in Las Vegas running several sports books - http://m.wired.com/magazine/2010/11/ff_midas/all/1. This has allowed them to set spreads for the entire NFL season (http://www.grantland.com/blog/the-triangle/post/_/id/27740/nfl-win-totals-hot-off-the-sportsbook-press) before the pre-season has taken place (which is not a typical case as most bookmakers seems to react on week to week action and player injuries and performance).

How are the actual odds generated? This is the more difficult question. Going on Mathletics (Wayne L.Winston 2009), some sports e.g. the NFL can be governed by a simple least squares algorithm based on margins of victory and points scored which can then be finessed (e.g. to give more weight to recent games). This can then be used to generate win percentages based on the ratings derived. In the case of the NFL, Hal Stern "On the Probability of Winning an American Football Game" (American Statistician 45, 1991) showed that the probability of the final margin of victory for home NFL team can be well approximated by a normal random variable with mean = home edge+home team rating-away team rating and a standard deviation of 13.86. Plug the ratings generated by your least squares work in and you have a set of percentages against a given spread. I believe that this can also be applied to a lot of other sports (e.g Australian Rules Football). In the case of football though I believe that oddsmakers also have done some regression analysis into player statistics to allow them to make a more rational rating based on the players that will actually be on the pitch rather than past team performance in terms of margins of victory (e.g. the Dtech group who analyse European football for the Times Newspaper base their ratings on the Team shots and goals data http://www.dectech.org/football/help_info.php - rather than a least squares model based purely on margin of victory). Given that sports could and should be viewed as an academic subject, I believe this is why we have seen increases in the number of the groups such as the Accuscore group who have a largely academic background (from interviews on the ESPN Behind the Bets Podcast) and have used their knowledge to generate opportunities from odds skewed to exploit gamblers that bet with pre-conditioned biases (e.g. the home team favourite wins more than 50% of games). If you can remove bias from the team that you pick, I believe this will generate opportunity.

Solved – Does martingale model work for betting football matches

We can put a probability on Barcelona winning at least once before you run out of money to fund the martingale strategy:

$ 10^6 = 2^i$ therefore $i = log_2(10^6) = 19.93157 \approx 19$

So you have 19 bets before you run out of cash.

Your profit percentage (given fractional odds) is $P=(OS_i - S_{i-1})/S_i$, where $i$ is the bet number you win on, $S_i$ is your stake, and $O$ are the odds (assumed constant). Since stake is given by $S_i = 2^{i-1}$ your percentage profit halves for every additional bet you have to take.

As time goes on, you're taking less and less rewarding bets. If bets are assumed independent from each other, your probability of winning nothing until the 19th bet are $p(1-p)^{18}/19$, which for $p=0.5$ is close to zero. However, your profit percentage would be roughly $1/10^6$ which is also tiny. It would correspond roughly to taking a bet at fractional odds of 1.000002 and betting a million dollars. Given that betting odds are not actually fair, if you took this bet enough times you'd lose money overall. House wins again :) Further, it's far worse than your return if you just put a million dollars in the bank.

Hope that helps.

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Solved – How do betting houses determine betting odds for sports

Solved – Does martingale model work for betting football matches

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