Theoretical Probability
Statistics & ProbabilityTheoretical 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. It assumes all outcomes are equally likely.
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.
Key Insight
Theoretical probability is your prediction before the experiment. Experimental probability is what actually happens when you run the experiment.
Definition
Theoretical probability 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. It is based on mathematical reasoning, not data collection.
Example
Rolling a fair die: $P(\text{prime number}) = P(\{2,3,5\}) = 3/6 = 1/2$. This is theoretical because we reason from the structure of the die, not from rolling it repeatedly.
Key Insight
Theoretical probability assumes a perfect, idealized model (e.g., a perfectly fair coin or die). Real objects may deviate slightly from the theoretical model, which is why experimental probability sometimes differs.
Definition
Theoretical probability is derived from a probability model without reference to observed data. It is the classical (Laplacian) definition for finite equally likely outcomes: $P(A) = |A|/|\Omega|$. For continuous distributions, it is derived from the assumed probability density function, often motivated by symmetry, maximum entropy, or physical reasoning.
Example
The probability that a randomly chosen real number from $[0,1]$ is irrational is $1$ by the Lebesgue measure argument: the rationals form a countable set with measure $0$. This purely theoretical result illustrates that theoretical probability can yield counterintuitive but mathematically rigorous conclusions.
Key Insight
Model specification is the key challenge in theoretical probability: the chosen model (uniform, normal, Poisson, etc.) embeds assumptions about the data generating process. Misspecified models yield incorrect theoretical probabilities, which is why model checking against experimental data is essential.