PRIORI PROBABILITY

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Priori Probability

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◇Overview

A Priori (Classical) Probability assigns probabilities before any empirical trials are performed, by deductive reasoning from the physical or mathematical symmetry of the experiment (such as fair coins, balanced dice, well-shuffled decks, or cryptographic uniform random number generators).

◇Key Definitions

• A Priori Probability Formula: When symmetry guarantees that all $N = |S|$ elementary outcomes in sample space $S$ are equally likely, the a priori probability of event $E \subseteq S$ is:
  $P(E) = \frac{\text{Number of outcomes favorable to } E}{\text{Total number of equiprobable outcomes in } S} = \frac{n(E)}{n(S)}$
• Contrast with A Posteriori (Empirical / Relative Frequency) Probability: A priori probability is deduced prior to experience from structural symmetry, whereas empirical probability $\hat{P}(E) = \frac{f}{n}$ is estimated after observing $n$ repeated trials.
• Contrast with Bayesian Prior / Posterior: In Bayesian inference, the prior $P(H)$ is the initial probability before new data $D$, updated via Bayes' Rule to the posterior $P(H \mid D)$.

◇Worked Example

Problem: Before drawing any card from a well-shuffled $52$-card deck, determine the a priori probability of drawing a Face Card (Jack, Queen, or King).

Solution:

• Each of the $4$ suits has $3$ face cards, giving $n(E) = 4 \times 3 = 12$ favorable cards out of $n(S) = 52$ equally likely cards.
• $P(\text{Face Card}) = \frac{12}{52} = \frac{3}{13} \approx 0.2308$.

◇References & Navigation

• Previous Topic: ◇Relative Frequency Form
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• Related Notes: ◇Additive Rule, ◇Expected Event