degrees of belief help us quantify how much we believe some event \(A\) is more/less plausible than some event \(B\).

Let us take two statements:

- \(A\) Taylor gets Nobel Prize in Literature
- \(B\) Han shot first

For instance, if we want to express “I think its more likely that Taylor gets the prize than Han shot first”:

\begin{equation} A \succ B \end{equation}

## axioms of degrees of belief

### universal comparability

for two statements \(A, B\), only three states can exist:

- \(A \succ B\) (A more likely)
- \(A \prec B\) (B more likely)
- \(A \sim B\) (equally likely)

### transitivity

if \(A \succeq B\) and \(B \succeq C\), then \(A \succeq C\)

### language of probability

using this framework, we can then describe the events in terms of probability

- \(P(A) > P(B) \Leftrightarrow A \succ B\)
- \(P(A) = P(B) \Leftrightarrow A \sim B\)

See also axiom of probability