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Odds

Odd define the risk:reward ratio of a bet. Different parts of the world use different formats but in the end, they all mean the same thing: the amount of money you get back based on your stake.
The most common odds type is decimal (or European) odds. These odds are decimal number greater than 1, usually rounded to 2 decimal places. Odds of 1.8 mean your return will be 1.8x larger than your stake. So your profit is 0.8x your stake. The implied probability calculated using decimal odds is defined as
1/decimalOdds1/decimalOdds
, so the implied probability of the 1.8 bet is 55.6%.
American odds are defined as positive integers above 100 or negative integers less than -100. Positive numbers define bets which are under 50% implied probability and represent the profit if you staked $100. So odds of +200 mean you get 200% profit (300% return) on your stake. Negative odds represent the amount you need to stake to get a profit of $100. So -150 means you need to stake $150 to profit $100. (If you are not American and are reading this thinking, "that is an insane system," you are correct.)
On most betting venues, the odds you see are the odds you get if your bet wins. This means there are no fees on top. On traditional betting exchanges, you are charged a "commission" on the profit of any winning bets. On traditional sports books, the "fees" are charged within the odds themselves. This is called the "vig" or overround. As an example, if they were offering betting on a coin flip, the odds would not be heads - 2.0, tails - 2.0. They would offer something like 1.90/1.90. Maybe they took a big bet on one side and needed more action on the other so would adjust it to 1.95/1.85. The smaller the overround, the better it is for users. The overround of a market is defined as
ΣOutcomes1decimalOdds\Sigma_{Outcomes} \frac{1}{decimalOdds}
. This number should always be >1 (100%). The greater the number, the higher the overround. If the number comes to <1, then the market can be exploited for risk-free money (e.g. a coinflip with odds for heads and tails both at 2.1. Betting $10 on both gets $1 profit no matter which side it lands on).
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