Odds and odds ratios are an important measure of the absolute/relative chance of an event of interest happening, but their interpretation is sometimes a little tricky to master. In this short post, I’ll describe these concepts in a (hopefully) clear way.

The Packers would be 1.714 in decimal odds as a -140 favorite in American odds. For every $1 risked, you’re profiting 71.4 cents, plus your $1 back. The Vikings would be 2.2 in decimal odds as a +120 underdog, meaning for every $1 risked, you’re profiting $1.20 and getting your $1 back. Pittsburgh Steelers +4.5 vs. Cincinnati Bengals -4.5. Above is an football point spread. Pittsburgh is +4.5, with Cincinnati at -4.5, which means Pittsburgh is a 4.5-point underdog and Cincinnati is favored by 4.5 points. For a bet on Pittsburgh to win at +4.5, they can either win the game outright or not lose by an amount that surpasses 4.5. It reflects the amount of money bet on a horse; the more money that is invested, the shorter the odds. When horse racing odds are shown in the form of 7-2, 5-1, etc, it expresses the amount of profit to the amount invested. So odds of 7-2 mean that for every $2 invested, the punter gets $7 profit in return. Of course, betting the New York Yankees at -1.5 to bring down the odds from -190 to -110 isn’t too fun when they win 4-3 and you don’t cash a bet. Betting on the point spread is the most.

From probability to odds

Our starting point is that of using probability to express the chance that an event of interest occurs. So a probability of 0.1, or 10% risk, means that there is a 1 in 10 chance of the event occurring. The usual way of thinking about probability is that if we could repeat the experiment or process under consideration a large number of times, the fraction of experiments where the event occurs should be close to the probability (e.g. 0.1).

The odds of an event of interest occurring is defined by odds = p/(1-p) where p is the probability of the event occurring. So if p=0.1, the odds are equal to 0.1/0.9=0.111 (recurring). So here the probability (0.1) and the odds (0.111) are quite similar. Indeed whenever p is small, the probability and odds will be similar. This is because when p is small, 1-p is approximately 1, so that p/(1-p) is approximately equal to p.

But when p is not small, the probability and odds will generally be quite different. For example if p=0.5, we have odds=0.5/0.5=1. As p increases, the odds get larger and larger. For example, with p=0.99, odds=0.99/0.01=99.

Fractional odds and gambling

Particularly in the world of gambling, odds are sometimes expressed as fractions, in order to ease mental calculations. For example, odds of 9 to 1 against, said as “nine to one against”, and written as 9/1 or 9:1, means the event of interest will occur once for every 9 times that the event does not occur. That is in 10 times/replications, we expect the event of interest to happen once and the event not to happen in the other 9 times. Using odds to express probabilities is useful in a gambling setting because it readily allows one to calculate how much one would win – with odds of 9/1 you will win 9 for a bet of 1 (assuming your bet comes good!).

Odds ratios

In the statistics world odds ratios are frequently used to express the relative chance of an event happening under two different conditions. For example, in the context of a clinical trial comparing an existing treatment to a new treatment, we may compare the odds of experiencing a bad outcome if a patient takes the new treatment to the odds of a experiencing a bad outcome if a patient takes the existing treatment.

Suppose that the probability of a bad outcome is 0.2 if a patient takes the existing treatment, but that this is reduced to 0.1 if they take the new treatment. The odds of a bad outcome with the existing treatment is 0.2/0.8=0.25, while the odds on the new treatment are 0.1/0.9=0.111 (recurring). The odds ratio comparing the new treatment to the old treatment is then simply the correspond ratio of odds: (0.1/0.9) / (0.2/0.8) = 0.111 / 0.25 = 0.444 (recurring). This means that the odds of a bad outcome if a patient takes the new treatment are 0.444 that of the odds of a bad outcome if they take the existing treatment. The odds (and hence probability) of a bad outcome are reduced by taking the new treatment. We could also express the reduction by saying that the odds are reduced by approximately 56%, since the odds are reduced by a factor of 0.444.

Why odds ratios, and not risk/probability ratios?

People often (I think quite understandably) find odds, and consequently also an odds ratio, difficult to intuitively interpret. An alternative is to calculate risk or probability ratios. In the clinical trial example, the risk (read probability) ratio is simply the ratio of the probability of a bad outcome under the new treatment to the probability under the existing treatment, i.e. 0.1/0.2=0.5. This means the risk of a bad outcome with the new treatment is half that under the existing treatment, or alternatively the risk is reduced by a half. Intuitively the risk ratio is much easier to understand. So why do we use odds and odds ratios in statistics?

Logistic regression

Often we want to do more than just compare two groups in terms of the probability/risk/odds of an outcome. Specifically, we often are interested in fitting statistical models which describe how the chance of the event of interest occurring depends on a number of covariates or predictors. Such models can be fitted within the generalized linear model family. The most popular model is logistic regression, which uses the logit link function. This choice of link function means that the fitted model parameters are log odds ratios, which in software are usually exponentiated and reported as odds ratios. The logit link function is used because for a binary outcome it is the so called canonical link function, which without going into further details, means it has certain favourable properties. Consequently when fitting models for binary outcomes, if we use the default approach of logistic regression, the parameters we estimate are odds ratios.

An alternative to logistic regression is to use a log link regression model, which results in (log) risk ratio parameters. Unfortunately historically these have suffered from numerical issues when attempting to fit them to data (see here for a paper on this). However there is also a more fundamental issue with log link regression, in that the log link means that certain combinations of covariate values can lead to fitted probabilities outside of the (0,1) range.

Case control studies

In case control studies individuals are selected into the study with a probability which depends on whether they experienced the event of interest or not. They are particularly useful for studying diseases which occur rarely. A case control study might (attempt to) enroll all those who experience the event of interest in a given period of time, along with a number of ‘controls’, i.e. individuals who did not experience the event of interest. In a case control study the proportion of cases is under the investigator’s control, and in particular the proportion in the study is not representative of the incidence in the target population. As a consequence, one cannot estimate risk or risk ratios from case control studies, at least not without external additional information. However, it turns out that the odds ratio can still be validly estimated with a case control design, due to a certain symmetry property possessed by the odds ratio.

Rare outcomes

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When the event of interest is rare (i.e. the probability of it occurring is low), the odds and risk ratios are numerically quite similar. Thus in situations with rare outcomes an odds ratio can be interpreted as if it were a risk ratio, since they will be numerically similar. However, when the outcome is not rare, the two measures can be substantially different (see here, for example).

If you've never set foot in an actual sportsbook before or logged into an online sportsbook, the chances of you getting overwhelmed when you actually do is very high. In an actual Las Vegas sportsbook, there is typically a lot of commotion and the odds and lines are displayed on a massive digital board for everyone to see. When a novice sports bettor looks at the massive digital signage, they will see a bunch of numbers, both positive and negative, some two digits, some three digits. They also won't have a clue what any of it means. The same can be said for the online sportsbooks. It looks like a massive spreadsheet with negative and positive numbers beside each teams' name.

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The easiest way for me to describe what all these numbers mean to you is to define it as point spread betting. Point spread betting is the most popular way to bet on the NFL and NBA, and it is a way for a sportsbook to generate betting interest on both sides.

Linemakers who work for the sportsbooks must put out lines that will entice the 'favorite' bettors to lay the points and take the favorite or entice the underdog bettors to take the points with the underdog.

As an example, let's say you are looking to place your very first wager on the Super Bowl. You look at the matchup either online or at a Las Vegas sportsbook and this is what you see:

Kansas City +4.5 (-110)
Carolina -4.5 (-110)

How Do Point Spread Bets Work?

Using the example above, the linemakers have determined that the Carolina Panthers are 4.5-point favorites over the Kansas City Chiefs. The favorite team can also be referred to as the chalk . The favorite will always be represented by a negative (-) number, while the underdog will always be represented by a positive (+) number.

Based on the line above and which team you decide to bet on, the Panthers must win by five or more points in order for those with a Panthers (-4.5) ticket to be declared a winner. As long as the Panthers win by five or more points, the final score itself does not matter. A 10-0 win is just as much a winner as a 56-50 win.

However, if the Panthers were to win the game by four or less points, then all Panthers backers can toss their tickets in the trash. A 17-14 or 21-17 Panthers win would cash the tickets with Chiefs +4.5 on them. A Falcons outright win as 4.5-point underdogs would do the same.

What is the -110 Line?

The standard price to pay when betting on point spreads is (-110). This is the sportsbooks way of ensuring a profit no matter which side covers the spread. The extra 10 cents is also known as the 'juice' or 'vig' . Paying the extra 10 cents is like paying a tax or commission to the sportsbook for brokering the bet.

The -110 line means that in order for you to profit $100 you must wager $110. Some sportsbook offer something called 'reduced juice' , which means that you can still profit $100 but the risk is a few dollars less.

For example, if you see reduced lines such as -7.5 (-105) that means that you must risk $105 dollars in order to profit $100. If you see -7.5 (-102) then you must bet $102 in order to profit $100. It may not seem like a big deal at the time, but saving a few bucks each time over the course of the season can really help your bankroll.

Which brings me to my next point. If you are serious about getting into sports betting, it is vital to have more than one sportsbook to make a wager at. Shopping around for the best lines will help your bankroll and you will be able to turn a bigger profit. If you see a pair of sneakers for $110 at one store, and the exact same pair is $102.99 at another store - which store are you buying them from?

What is a Push?

When betting the point spread, there is almost always a winner and a loser. However, in some instances sportsbook decide to put out a whole number such as -3 for bettors to bet on. If the final score ends with a differential of three points - no matter who wins - the bet is considered a 'push' and all money is refunded to both sides since neither team covered the spread.

What does Pick'em or PK Mean?

When two teams are evenly matched, and the sportsbook can't decide which team should be the favorite, they will release PK lines which means neither team is favorite. The team you wager on must simply win the game by any score in order for your ticket to be graded as a winner.

What Happens When the Point Spread Changes?

This is a very common occurrence throughout the sports betting industry. Sportsbooks have the right to shift the spread or odds for any given match prior to it starting. Many factors play a huge role in this decision, and they include injuries, weather, the volume of bets on one side, and anything in between. Depending on the time you place your wager, the bettor may also have an advantage or disadvantage based on which way the spread has shifted.

For example:

Opening Line: Carolina -4.5 vs Atlanta +4.5

Thursday's Line: Carolina -2.5 vs Atlanta +2.5

If bettors were quick to jump on the Atlanta line at +4.5 when it first came out, they would have a distinct advantage over those who waited closer to kick off and were stuck with +2.5. The opposite holds true for Carolina. Bettors that were quick to pull the trigger are now laying two more points than they would if they were patient and saw the line movement before making their move.

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