Probabilities and odds express the same information in different ways. In science and economics, people tend to think in terms of probabilities, while in gambling, people tend to think in terms of odds.

Probabilities express the likelihood of an event occurring as a percentage of all possible events occurring. Odds express the number of times an event will occur with relation to the number of times the event will not occur.

Imagine we have five marbles in a bowl—we have three red marbles and two blue marbles, as shown below.



If we randomly select one marble, the probably of picking red can be expressed as:



Where:
Pr = Probability of picking a red marble
Nr = Number of red marbles
Nb = Number of blue marbles

In the above example, the equation evaluates as follows:



Thus, the probability of randomly picking a red marble is 3/5 or 0.6 or 60%.

The same information can be expressed as odds:



Where:
Or= Odds of picking a red marble

This evaluates as follows:



The odds of picking red can be expressed as 3:2. In the case of betting however, odds are typically expressed as the odds of losing, rather than the odds of winning. In other words, it is generally described as the odds against an event occurring. In this case, the odds of not picking red are 2:3. If, for example, the odds are expressed as 9:1 for a particular team, the interpretation is that the team will likely lose nine times for each one win.

It is also easy to convert from probabilities to odds and vice versa. The conversion of probability to odds is calculated as:



In the above example, this evaluates as:



If we return to the original calculation above, the odds of picking red are expressed as 3/2, which is equivalent to 1.5 or 1.5:1.

We can therefore convert from odds to probability as:



People often use the terms ‘probability’ and ‘odds’ interchangeably. While these concepts ultimately can be used to describe the same phenomenon, they have distinct mathematical formulations.