# Probability - Odds (Ratio)

The odds in favor of an event is the ratio of the probability that the event will happen to the probability that the event will not happen.

An odds ratio is just the probability of an event (outcome).

$$\begin{array}{rrl} \text{Odds} & = & \frac{\text{Probability that the event will happen}}{\text{Probability that the event will NOT happen}} \\ & = & \frac{\text{Probability that the event will happen}}{1 - \text{Probability that the event will happen}} \\ & = & \frac{\text{Probability of the outcome}}{\text{Opposite probability of the outcome}} \\ \end{array}$$

or $$\begin{array}{rrl} \text{Probability that the event will happen} & = & \frac{\text{Odds}}{1-\text{Odds}} \end{array}$$

## Example

For example, the odds that a randomly chosen day of the week is a Sunday are: $$\begin{array}{rrl} \text{Odds} & = & \frac{ \frac{1}{7} }{ \frac{6}{7} } \\ & = & \frac{ 1 }{ 6 } \\ \end{array}$$

## against/in favor

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

## Logit

The natural log of the odds is call the log-odds or logit.

The logit will always fall between 0 and 1.