Risks and Odds (v2)
Risks and Odds Are Not The Same
1 Background
Risks and Odds are different but related quantities. It it important to understand how each is mathematically defined and to understand what each quantity implies. In some cases, the odds may overstate the risk.
2 Risk
Let’s think about risk:
\[\text{risk} = P(\text{event happened}) = \frac{\text{number of events}}{\text{number of events} + \text{number of non-events}}\]
3 Odds
The odds are the probability that an event happened divided by the probability that it did not happen
\[\text{odds} = \frac{P(\text{event happened})}{P(\text{event didn't happen})}\]
which in turn is equivalent to
\[= \frac{\frac{\text{number of events}}{\text{number of events} + \text{number of non-events}}}{\frac{\text{number of non-events}}{\text{number of events} + \text{number of non-events}}}\]
which incidentally reduces to
\[= \frac{\text{number of events}}{\text{number of non-events}}\]
4 Visualize The Risk And Odds
Imagine an Event X. This event could be entering a program, exiting a program, or getting a diagnosis of a mental health or physical health condition.
Click on the table below and/or hover over the graph below to explore various scenarios.