This is a pretty good explanation of odds and probabilities, from Wikipedia:
The odds in favor of an event or a proposition is the ratio of the probability that the event will happen to the probability that the event will not happen. For example, the odds that a randomly chosen day of the week is a Sunday are one to six, which is sometimes written 1 : 6.
'Odds' are an expression of relative probabilities. Often 'odds' are quoted as odds against, rather than as odds in favor. For example, the probability that a random day is a Sunday is one-seventh (1/7), hence the odds that a random day is a Sunday are 1 : 6. The odds against a random day being a Sunday are 6 : 1. The first figure represents the number of ways of failing to achieve the outcome and the second figure is the number of ways of achieving a favorable outcome.
In probability theory and Bayesian statistics, odds may sometimes be more natural or more convenient than probabilities. This is often the case in problems of sequential decision making as for instance in problems of how to stop (online) on a last specific event which is solved by the odds algorithm.
Stating "odds against" is a convenient way to propose a bet. When a bookmaker offers betting odds of 6 : 1 against some event occurring, it means that he is prepared to pay out a prize of six times the stake, and return the stake as well, to anyone who places a bet, by making the stake, that the event will occur. If the event does not occur, then the bookmaker keeps the stake. For example, a winning bet of 10 at 6 : 1 against will win '6 × 10 = 60' with the original 10 stake also being returned. Betting odds are skewed to ensure that the bookmaker makes a profit — if true odds were offered the bookmaker would break even in the long run — so the numbers do not represent the bookmaker's true odds.
"Odds on" means that the event is more likely to happen than not. This is sometimes expressed with the smaller number first (1 : 2) but more often using the word "on" (2 : 1 on) meaning that the event is twice as likely to happen as not.
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In casino games odds are always expressed as odds againt. If lotteries were more honest they'd be expressed that way too, but it's usually the reverse, instead of 175,000,000:1 it's expressed as 1:175,000,000. Yeah, it's the same thing expressed in different ways, but with a marketing twist.
A lot of people mix up possibilities for probabilities. In roulette, there are three possible colors that can result, black, red, or green (0,00) but if you didn't bet the 0 or 00 you don't have green covered.
On a crap table there is only one way a 12 can be rolled, a 6 and a 6. There are 36 possible combinations with a pair of dice. So the probability of rolling a 12 is 1 in 36, or 35:1 (35 to 1). But if somone bets it and it hits the payoff is 29 to 1 or 30 to 1, depending on the casino. Thus the house edge.
Sone layouts were labeled 31 for 1 to attract business but when the word for is used it includes the bet.