Odds

In probability theory, odds provide a measure of the probability of a particular outcome. Odds are commonly used in gambling and statistics. For example for an event that is 40% probable, one could say that the odds are "2 in 5", "2 to 3 in favor", or "3 to 2 against".

When gambling, odds are often given as the ratio of the possible net profit to the possible net loss. However in many situations, you pay the possible loss ("stake" or "wager") up front and, if you win, you are paid the net win plus you also get your stake returned. So wagering 2 at "3 to 2", pays out 3 + 2 = 5, which is called "5 for 2". When Moneyline odds are quoted as a positive number $+ X$ , it means that a wager pays $X$ to 100. When Moneyline odds are quoted as a negative number $- X$ , it means that a wager pays 100 to $X$ .

Odds have a simple relationship with probability. When probability is expressed as a number between 0 and 1, the relationships between probability $p$ and odds are as follows. Note that if probability is to be expressed as a percentage these probability values should be multiplied by 100%.

" $X$ in $Y$ " means that the probability is $p = X / Y$ .
" $X$ to $Y$ in favor" means that the probability is $p = X / (X + Y)$ .
" $X$ to $Y$ against" means that the probability is $p = Y / (X + Y)$ .
"pays $X$ to $Y$ " means that the bet is a fair bet if the probability is $p = Y / (X + Y)$ .
"pays $X$ for $Y$ " means that the bet is a fair bet if the probability is $p = Y / X$ .
"pays $+ X$ " means that the bet is fair if the probability is $p = 100 / (X + 100)$ .
"pays $- X$ " means that the bet is fair if the probability is $p = X / (X + 100)$ .

The numbers for odds can be scaled. If $k$ is any positive number then $X$ to $Y$ is the same as $kX$ to $kY$ , and similarly if "to" is replaced with "in" or "for". For example, "3 to 2 against" is the same as both "1.5 to 1 against" and "6 to 4 against".

When the value of the probability $p$ (between 0 and 1; not a percentage) can be written as a fraction $N / D$ then the odds can be said to be " $p /(1- p)$ to 1 in favor", " $(1- p)/ p$ to 1 against", " $N$ in $D$ ", " $N$ to $D - N$ in favor", or " $D - N$ to $N$ against", and these can be scaled to equivalent odds. Similarly, fair betting odds can be expressed as " $(1- p)/ p$ to 1", " $1/ p$ for 1", "+ $100(1- p)/ p$ ", " $-100 p /(1- p)$ ", " $D - N$ to $N$ ", " $D$ for $N$ ", "+ $100(D - N)/ N$ ", or " $-100 N /(D - N)$ ".