Theoretical Probability
Theoretical probability is the expected probability of an event calculated from mathematical reasoning before any experiment is conducted.
Formula
\text{P(event)} = \dfrac{\text{favorable outcomes}}{\text{total equally likely outcomes}}
Definition
Theoretical probability is what you expect to happen based on math and logic, without actually running an experiment, assuming all outcomes are equally likely. It uses the ratio of favorable outcomes to total equally likely outcomes, $P(A) = |A|/|S|$, where $|A|$ is the number of outcomes in event $A$ and $|S|$ is the total number of outcomes in the sample space, based on mathematical reasoning rather than data collection. Formally, theoretical probability is derived from a probability model without reference to observed data: it is the classical (Laplacian) definition for finite equally likely outcomes, $P(A) = |A|/|\Omega|$, and for continuous distributions it is derived from the assumed probability density function, often motivated by symmetry, maximum entropy, or physical reasoning.
Example
Before flipping a fair coin even once, you know theoretically that $P(\text{heads}) = 1/2$, because there is $1$ heads out of $2$ equally likely outcomes. Rolling a fair die, $P(\text{prime number}) = P(\{2,3,5\}) = 3/6 = 1/2$, reasoned from the structure of the die rather than from rolling it repeatedly. The probability that a randomly chosen real number from $[0,1]$ is irrational is $1$ by the Lebesgue measure argument, since the rationals form a countable set with measure $0$, a purely theoretical result illustrating that theoretical probability can yield counterintuitive but mathematically rigorous conclusions.
Key Insight
Theoretical probability is your prediction before the experiment, while experimental probability is what actually happens when you run it; theoretical probability assumes a perfect, idealized model (a perfectly fair coin or die), and real objects may deviate slightly, which is why experimental probability sometimes differs. Model specification is the key challenge in theoretical probability: the chosen model (uniform, normal, Poisson, etc.) embeds assumptions about the data generating process, and misspecified models yield incorrect theoretical probabilities, which is why model checking against experimental data is essential.