I am having some trouble in understanding odds, and I would like just a basic explanation for how to interpret them.

I have found various posts related to odds but most of them are more complex than what I am trying to understand. Here is an example of how I am interpreting odds: if the odds of an event happening are 3 to 1, then the event will happen 3 times for every 1 time that it doesn't happen. I do not know if this interpretation would be correct. So, any guidance and more examples on interpreting odds would be greatly appreciated.

On another thread there is a much broader answer by @gung that also deals with related technical issues such as the odds ratio, but I am going to stick to the topic at hand: how to interpret odds, and particularly the formulation "$a$ to $b$". As a beginner's question, it's worth thinking how "odds" are expressed in everyday speech (especially in betting parlance) as well as what odds mean to a statistician, because discrepancies between the two are problematic for learners.

For odds as expressed by a statistician, your contention is correct. Suppose a bag contains four tokens, of which three are $\color{aquamarine}{\text{aquamarine}}$ and one is $\color{brown}{\text{brown}}$, and one token is selected at random. The probability that the selected token is aquamarine is 3 out of 4, i.e. $\frac{3}{4}$, often read "3 in 4". With equally likely outcomes, the odds for aquamarine would be calculated as the number of favourable outcomes (3) divided by the number of unfavourable outcomes (1), which is $\frac{3}{1} = 3$, often read as $\color{aquamarine}{\text{three}}\text{ to }\color{brown}{\text{one}}$ or just as the number "three". More generally, you could take the fraction of "favourable outcomes over unfavourable outcomes" and cancel (divide) both numerator and denominator by the total number of outcomes, to obtain "the probability of a favourable outcome over the probability of an unfavourable outcome", from which a little algebra gives:

$$\text{Odds} = \frac{p}{1-p} \implies p = \frac{\text{Odds}}{1 + \text{Odds}}$$

Odds as expressed by a bookmaker are usually quoted either as "odds against" or "odds on", and which way they are written seems to be a common cause of confusion. In so-called British odds, fractional odds or traditional odds, the odds for aquamarine would be written "3/1 on" or "3-1 on", read as $\color{aquamarine}{\text{three}}\text{ to }\color{brown}{\text{one}}\text{ on}$.* For a gambler, the fact these are "odds on" indicates that a stake of £3 on aquamarine would return £1 profit if successful (they actually receive £4, of which £3 is simply the return of the original stake) whereas a failed bet results in the loss of the £3 stake. We can see these are "fair odds" because the gambler has three chances of gaining £1 and one chance of losing £3, so on average there is no expected gain or loss. So far, so little discrepancy: "odds on" are simply the "odds in favour" preferred by statisticians.

For events with a 50% probability, such as heads on a coin toss — two equally likely outcomes of success or failure — a statistician would say the odds are "one to one", $\frac{1}{1}$ or simply $1$ whereas a fair bookmaker would give fractional odds of 1/1 (read as "evens"). So no problems here either; however, when the probability falls below 50%, we see the bookmaker resumes quoting the larger number in the ratio before the smaller one.

Consider a race in which all four horses (let's say Foinavon, Gregalach, Mon Mome and Tipperary Tim) are equally likely to win: then in terms of probabilities, we would say each had a "1 in 4" or 0.25 chance of victory. What would the fair odds be for a bet on, say, Foinavon? There is only one favourable outcome (victory for F) versus three unfavourable outcomes (victory for G, M or T), so a statistician would describe the odds as "1 to 3", or numerically as $\frac{1}{3}$. However, a bookmaker using British odds would see the odds as "3 to 1 against", and write them as simply "3/1" or 3-1" (both read "three to one"; the "against" is implicit and goes unspoken). For a gambler, "odds against" means a stake of £1 would return £3 profit if successful (they will actually receive £4, but £1 of this is the return of the original stake) whereas if unsuccessful they lose the £1 stake. The gambler has three chances of losing £1 and one chance of gaining £3, so again there is zero expected profit/loss and the odds are fair. Sadly, "odds against" (the usual form of odds) does not correspond to a statistician's "odds in favour".

Each horse in our hypothetical race achieved fame by once winning the Grand National at odds of 100/1: since these were high ("long") odds against, they were "long shots" considered extremely unlikely to win, and their backers were handsomely rewarded with £100 profit per pound wagered. If we pretend that the bookmakers' odds were the fair odds (which would ignore the bookmaker's overround, or "vig"), it was felt that there were 100 ways the horse could lose for each way the horse could win, so the implied probability of success was $\frac{1}{101}$. On the contrary, if a statistician claimed an event had odds of "100 to 1", that is a claim the event is very likely (with a probability of $\frac{100}{101}$).

If any layperson in your audience comes from a country where fractional odds are used by bookmakers, and regularly quoted in the media (e.g. "Jeremy Corbyn set to beat 100-1 odds to become leader of UK's Labour party", The Guardian, 11 September 2015; "11 million to one: Quadruplet calves born in South Australia", Sydney Morning Herald, 30 July 2015) then quoting odds in the form "$a$ to $b$" is almost certain to cause confusion.

I've seen people try this, perhaps in the belief that "the general public is more familiar with odds than probabilities", but statisticians wise to the bookmaker's overround, and who have therefore never placed a bet in their lives, may be caught by surprise that the popular conception of odds is "the wrong way round". If this confusion is felt to outweigh the advantages of the "$a$ to $b$" formulation (particularly that it makes clear odds express a ratio of favourable to unfavourable) then it might be better to express "statistical odds" as a single number, to distinguish them from a bookmaker's fractional odds. Before presenting statistical odds to such an audience, I would at least make them aware of the following points:

• A statistician's odds correspond to a bookmaker's "odds on". If you are used to "odds against", a statistician's odds may seem "the wrong way round". For example, "10 to 1" indicates a very likely event, and "1000 to 1" an extremely likely one!
• A statistician need not put the higher number first, so odds like "2 to 3" can be used to indicate "2 chances of success to 3 chances of failure" (i.e. after many trials, the ratio of successes to failures should be around 2:3 and hence the probability of success is $\frac{2}{5}$).
• While bookmakers prefer to give odds as a ratio of whole numbers,** statisticians will often simplify their odds into the form "something to one", even if this introduces a decimal (e.g. "5 to 2" becomes "2.5 to 1").
• A statistician may leave off the "to one" and quote a single number (e.g. odds of 3.5 mean "3.5 to 1", or "7 to 2", so the long-run ratio of successes to failures is expected to be 7:2, from which the probability of success can easily be seen to be $\frac{7}{9}$).
• On this scale, odds of zero represent an impossibility; odds between 0 and 1 indicate a less-than-evens chance; odds of 1 show a 50% chance; odds above 1 indicate the event is more likely than not; a certain event would have infinite odds.

Mathematically, we have

$$\text{Odds}_\text{statistician} = \text{Odds on}_\text{British}; \quad \text{Odds}_\text{statistician} = \frac{1}{\text{Odds against}_\text{British}}$$

Even this may not be sufficient to avoid confusion. Decimal odds, also known as continental odds or European odds, have become more prevalent in an era of online gambling, especially for in-play betting and betting exchanges where fractional odds are unwieldy for displaying small, rapid changes in implied probabilities. European odds quote the payout per unit staked, including the return of the stake. For the aquamarine bet, a winning £3 stake makes a profit of £1, so each £1 staked would make a profit around £0.33 (in a payout of £1.33). The European odds for aquamarine are therefore about $1.33$. For the coin toss, a gambler staking £1 receives a payout of £2 (if successful) or £0 otherwise, so the European odds are $2.00$. For a £1 bet on Foinavon, a gambler has a winning payout of £4, so the European odds are $4.00$. You may have noticed that the European odds are the reciprocal of the implied probability of success: for the odds to be fair on a £1 stake, the expected payout (which is the probability of success multiplied by the winning payout) must equal the £1 wagered, so the winning payout must be the reciprocal of the probability. Since $\text{Odds}_\text{European} = \frac{1}{p}$ we find

$$\text{Odds}_\text{statistician} = \frac{p}{1-p} = \frac{1}{p^{-1}-1} = \frac{1}{\text{Odds}_\text{European}-1}$$

We might also have deduced this from noting $\text{Odds}_\text{European} = \text{Odds against}_\text{British} + 1$ (because of European odds including the return of the stake in the payout).

European odds have several advantages to the gambler. Comparing two fractional odds (try 8/15 versus 4/7) involves greater feats of mental arithmetic than comparing two decimals. Small changes to the implied probability work "smoothly" for a decimal whereas the form of a fraction may have to completely change as a different denominator is required. Calculating the payout from a win is as simple as multiplying the stake by the European odds (e.g. a winning stake of €300 at European odds of $1.50$ receives a payout of €450, of which €150 is profit). The reciprocal relationship with implied probability is especially useful for spotting "value bets": if a gambler believes the true probability of success on a bet at European odds of $6.00$ is greater than the bet's implied probability of $\frac{1}{6}$, the bet is good value and the gambler's expected profit is positive.

However, it's harder for a statistician to explain mathematical odds to a layperson accustomed to European odds! Like British "odds against", higher European odds indicate an event that's deemed less likely ($1.00$ for a certainty, $2.00$ for an even chance, $\infty$ for an impossibility). Even worse, the numbers are not simply the "wrong way round" but completely misleading: the entire concept of a ratio of favourable and unfavourable outcomes has been lost.

This key conceptual ratio is retained in the moneyline system used in US sports betting, even though it looks more complex at first sight. Positive figures indicate profit (excludes return of stake) on a winning $\$100$stake, essentially the same idea as "odds against". A figure of +300 indicates$\$300$ of profit on a $\$100$stake, equivalent to "3/1 [against]" in the British system or "1 to 3" for a statistician (the Foinavon bet). Negative figures indicate the required stake to win a profit of$\$100$, equivalent to "odds on". A figure of -300 shows a $\$300$stake makes$\$100$ profit, which is "3/1 on" in the British system or "3 to 1" for a statistician (the aquamarine bet).

$$\text{Odds}_\text{statistician} = \begin{cases} \frac{|\text{Moneyline}|}{100} & \text{if Moneyline} < 0 \\[5pt] \frac{100}{\text{Moneyline}} &\text{if Moneyline} > 0 \end{cases}$$

I appreciate much of this answer has concerned betting and pay-offs rather than statistics, but I've found the everyday usage of "odds" differs so markedly from the statistician's technical definition, that a thorough comparison might address some confusion (both of non-technical gamblers, and non-gambling statisticians). There are, of course, deep historical and philosophical links between betting and statistics. The problem of points concerned the fair division of the prize pot in an interrupted gambling game, and had generated discussion since medieval times. When Antoine Gombaud, chevalier de Méré posed a version of the problem in 1654, the subsequent correspondence of Blaise Pascal and Pierre de Fermat on the issue laid the foundations of probability theory. More recently, Frank Ramsey (in the 1920s) and Bruno de Finetti (in the 1930s) examined the coherence of wagers (related to the gambling phenomenon of a Dutch book) as a justification of Bayesian probability: if an agent's subjective probabilities or degrees of belief do not obey the axioms of probability, then they are incoherent and a Dutch book can be made against the agent, exposing them to a certain loss. The Stanford Encyclopedia of Philosophy has an article on the "Dutch Book argument".

($*$) I've deliberately oversimplified here for pedagogical purposes. In fact bookmakers are not consistent on this point: these odds may well be written "1/3" (signifying "one to three against"), though this may still be read aloud as "three to one on"! However, while a bookmaker might write the smaller number first in an odds against bet, they will never frame an odds on bet in this way: "1/3 on" would theoretically be the same as "3/1 [against]", but in practice would always be quoted in the latter form.

($**$) As an aside, bookmakers do not always cancel these whole numbers to their lowest terms: "6/4" is often advertised ("ear'ole"), so perhaps bookmakers believe a £6 profit on a £4 stake is more psychologically enticing than the prospect of £3 profit on a £2 stake. I have heard it argued, though the truth I know not, that "100/30" survives because "10 to 3" could be mistaken for the time of a race. Hong Kong odds are fractional odds (against) cancelled down to a single number, so "5/2 against" becomes 2.5; the profit from a winning bet (excluding return of the stake) is then the Hong Kong odds multiplied by the stake. Hong Kong odds below one indicate a greater than 50% chance; they are the reciprocal of statistical odds.

• When I was 14 and studied statistics as a separate subject at high school for the first time, my textbook examined carefully gambling odds and payoffs versus probabilities and "statistical odds": in retrospect, the level of detail was rather disturbing :) Completists may mourn the absence of Caughoo's controversial 1947 Grand National victory, the only other 100/1 winner, but in keeping with the original question I wanted to compare "1 to 3" and "3 to 1", leaving no room for Caughoo in the lineup. – Silverfish Nov 27 '15 at 14:13
• I'm not sure is gung's answer is really "much broader" right now ;) – Tim Nov 27 '15 at 15:43
• A much more in-depth answer than I thought it would be. +1 – Jessica Dec 2 '15 at 19:26