Naïve Definition of Probability

The "naïve definition of probability" states that to calculate the probability of an event you count the number of ways that the event could occur and divide it by the number of all possible outcomes.

$\textbf{Definition}$ (Naïve Definition of Probability). Let $A$ be an event for an experiment with a finite sample space $S$. The $\textit{naïve probability}$ of $A$ is: $$P_{naïve}\left(A\right) = \frac{|A|}{|S|} = \frac{\text{number of outcomes favourable to $\textit{A}$}}{\text{total number of outcomes in $\textit{S}$}}$$ Where $|A|$ denotes the cardinality of set $A$. [1]

Using this definition only works when all outcomes are equally likely by design or there is symmetry present in the problem that makes them such. Examples include tossing a fair coin, drawing from a well-shuffled deck of cards, selecting $n$ people from a group where everyone is equally likely to be selected, etc.


[1]    J. K. Blitzstein and J. Hwang, Introduction to Probability /, Second edition. (2019).