Area under curve below cutoff: 0.0500Probability vs Likelihood vs Odds
Let’s first compare between probability and likelihood. Let’s say we have data from a normal distribution with mean = 30 and standard deviation = 1. The probability of getting less than the observed value 28 given the distribution is \(P(X<=28~|~N(30, 1))\) which is the area under the curve below 28 as shown in the graph.
The likelihood of a distribution with mean = 30 and standard deviation = 1 given an observation = 28 is the y-axis value of our normal probability density function from the observed data point i.e., \(L(N(30, 1)~|~X=28)\).
Simply put, a probability is the area under the curve in a fixed distribution and mathematically we say:
\[P(Data~|~Distribution)\]
A likelihood is a y-axis value for a fixed data point with distributions that can be moved and mathematically we say:
\[P(Distribution~|~Data)\]
Now let’s compare probability with odds. Odds are the ratio of an event happening it to it not happening:
\[\frac{\text{event } \textcolor{red}{\text{happening}}}{\text{event not } \textcolor{red}{\text{happening}}}\]
Whereas probability is the ratio of an event happening to all possible events:
\[\frac{\text{event } \textcolor{red}{\text{happening}}}{\text{all possible events}}\]