Probability Review I

Notions of Probability

Let us look at two notions of the probability of an event happening:

The Frequentist Version

Given $N$ distinct possible events, $\left(x_{1},x_{2},\ldots ,x_{N}\right)$, with the following assumptions:

• No two events can occur simultaneously, and
• The events occur with frequencies $\left(n_{1},n_{2},\ldots ,n_{N}\right)$,

Then the probability of an event $x_{i}$ is given by:

$P\left(x_{i}\right)={\frac {n_{i}}{\sum _{j=1}^{N}n_{j}}}$

(1)

From this definition, we can see that:

$\sum _{i=1}^{N}P\left(x_{i}\right)=1$

(2)

Observer Relative Probability

Probability is an assertion about the belief that a specific observer has of the occurrence of a specific event. Thus, here, two different observers may assign different probabilities to the same event or phenomenon. Additionally, the probability assigned to an event is likely to change as the observer learns more about an event, or if certain aspects of the event changes.