Dec 8, 2022
Dec 8, 2022

Beating the vigorish with conditional probability in sports betting

What is conditional probability?

What is a sports betting market? The most basic definition is that a betting market is where money is exchanged between bettors and bookmakers on the outcome of a particular event, with prices set and moved according to a variety of factors.

What is Vig (Vigorish)?

A betting market will have a margin applied to it which provides commission to the bookmaker (essentially the charge for taking the bet) and this is also referred to as the vigorish, vig or juice. As a bettor, we not only have to beat all the other players in the market, but we also must find enough of an edge on the selections we bet on to overcome this vig.

How do we beat the vigorish in betting markets?

To find an edge in the sports betting markets we need to look for angles and variables that are not already factored into the price (odds). We can do this using various methods of compiling our own odds, or we can monitor the movement of the odds through weight of money or sharp bettors’ opinions. The buzzword for the latter concept is “Steam Chasing”.

Most betting markets or sophisticated bettors’ tissue odds are formed using mathematical models nowadays. A lot of the output prices will be derived from what are known as frequentist probabilities and big-data averages.

The aim for the bookmaker is to beat the average of all punters.

If we take each market and selection in a vacuum, what usually isn't considered in the pricing up of a market is conditional probability and this is where we, as value bettors, can strike.

Conditional probability is the measure of the probability of an event happening given that another event that affects the percentages has occurred. Bookmakers don't have the time, the resources, or the need to base the odds on Bayesian inference which includes these situational factors.

The aim for the bookmaker is to beat the average of all punters and they have a head start given the vig applied to the market, and the potential factoring of sharp bettors’ accounts.

The mathematical concepts behind this pricing can be described as heuristics, which are rules of thumb for pricing universal markets and means it can be profitable for both the bookmaker and the sports bettor. I refer to the betting markets and how they can work for both bookmakers and bettors as the Blackbox Paradox.

What is the vig in sports betting markets?

Vig is applied for profit and protection in the betting markets. Theoretically, moving prices on weight of money to balance the book covers liability. The standard way to add vig to a two-way market where both options are 50/50 is to cut the prices to 10/11 each side.

If we divide the denominator by the numerator + denominator and then multiply by 100 it will look like this:

11/21 = 0.524

0.524 * 100 = 52.4%.

So, 52.4% is the implied probability and if we add this up for both sides, we get 104.8%. The total is called the overround and the extra 4.8% is the vig (or margin or commission).

The vig, written as a percentage, is thus:

1 - (1/overround) * 100

For our example above, it will look like this:

1 – (1/104.8) * 100 = .0458 or 4.58%

The bookmaker will theoretically hold 4.58% commission/vig on that market.

How do we go about finding edges that can overcome the commission?

The first secret to finding an edge is to think logically. For example, in soccer, various in-play odds are derived using a method called time decay which doesn’t change for situational factors which might be live in certain matches.

And in golf, an in-play betting algorithm is not going to take into account dramatic changes in the weather, and in reality, different probabilities might come into play as a result of unexpected weather conditions. Also, we might have a course playing so long that half the field is at a significant disadvantage, which means the probabilities change but the odds often won’t reflect that.

Place markets in golf will be derivatives of outright prices but if a player is priced at 100.000 to win, it doesn’t necessarily mean he is 10.000 for a top-10 finish or 5.000 for a top-20 finish. We must consider factors like current form or course fit for each specific player.

My book Hypnotised by Numbers looks at the following theoretical example in golf betting:

The average hole out percentage for an elite professional golfer is around 50% from eight feet. If we could choose to bet on each golfer individually to hole from this range, the probabilities can change. Ask yourself:

• What quality of putter is the player?
• What type of grass are they putting on?
• How will the slope and speed of the surface change the difficulty?

If betting on this type of prop bet with the advantage of being selective and with strong knowledge of the player’s putting tendencies, we could expect to beat the odds.

A strike rate of around 60% is achievable versus the 52.4% needed to break even against the vig. We could pick strong putters with putts on greens running at 11-12 on the stimpmeter. Doing this would see the hole out percentage from eight foot be something closer to 60%. The 50% figure is just an average over the whole population.

This is the power of selectivity for sports bettors. We now have a 60% win-rate on a 10/11 shot (with vig applied) through conditional probability or pure logic. This equates to a return on investment of 14% in the long term, even against the vig on a true even-money shot.

What happens in these types of prop markets is the bookies win against the average player and on the whole, while a small selection of smart players can also win. If we take every match, game, market, and sport individually there will be many opportunities where we can pick up on mispriced odds and take advantage, as certain nuances of the sports themselves won’t always be priced into the betting markets.

In Hypnotised By Numbers, we also looked at the example of Denzel Dumfries for the Netherlands team when touted as a right back in the Euro 2020 matches. He was priced up from available data at around 10/1 to score anytime which is accurate for an attacking full-back.

However, Denzel played more as a forward player on the flank. He scored in the opener, and nothing changed for Matchday 2 where he was priced up the same. Dumfries scored again. A little knowledge is a dangerous thing. He also got man-of-the-match in both games at big prices of 21.000.

If you examined soccer prop betting odds using conditional probability techniques and watched that first match, using Bayesian reasoning you could have deduced that the fair odds for Denzel to score anytime were significantly less than 11.000.

Beating the vig with conditional probability - a summary

Odds are implied probabilities and emphasis should be on the word implied. The market often isn’t as efficient as it seems because even the biggest players are all using the same data - data that doesn't take into consideration situational factors. Odds, models, and algorithms are based around frequentist statistics. Applying conditional probability is how we can beat them significantly, overcoming the vigorish in the process. Successful sports betting is about understanding the fundamental mathematics behind the markets and applying logic to beat them. We don’t want to fight fire with fire by using the same methods of pricing and the same data as the odds compilers.

There’s a cognitive bias known as the Dunning-Kruger Effect which can play a part in the Efficient Market Hypothesis so don’t be too quick to think the markets can’t be beaten. When floods of money come into the betting markets and change the odds, this is actually Bayesian Inference in disguise. We can apply conditional probability to the sports betting markets that are priced on big data and get more accurate estimates of true prices. There is plenty of value to find out there, it’s how you apply your logic that separates your winning bets from losing ones.