BASIC FORMULA OF PROBABILITY

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Basic Formula of Probability

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◇Overview

The Classical (Laplacian) Formula of Probability defines the probability of an event $E$ in a finite sample space $S$ of equally likely outcomes as the ratio of the number of outcomes favorable to $E$ to the total number of outcomes in $S$, grounded in Kolmogorov's Three Axioms.

◇Key Definitions

• Classical Probability Formula: If a sample space $S$ consists of $N = |S|$ equally likely outcomes and event $E \subseteq S$ contains $n(E) = |E|$ outcomes, then $P(E) = \frac{n(E)}{n(S)} = \frac{|E|}{|S|}$.
• Axiom 1 (Non-negativity): $P(E) \ge 0$ for every event $E \subseteq S$.
• Axiom 2 (Normalization): $P(S) = 1$ for the entire sample space $S$.
• Axiom 3 (Countable Additivity): For any pairwise disjoint events $E_1, E_2, \dots$, $P\left(\bigcup_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty P(E_i)$.

◇Worked Example

Problem: Two fair six-sided dice are rolled. What is the probability that the sum of the two upward faces equals $8$?

Solution:

• Total equally likely ordered pairs in $S$: $n(S) = 6 \times 6 = 36$.
• Favorable outcomes summing to $8$: $E = \{(2,6), (3,5), (4,4), (5,3), (6,2)\}$, so $n(E) = 5$.
• $P(E) = \frac{5}{36} \approx 0.1389$.

◇References & Navigation

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• Related Notes: ◇Relative Frequency Form, ◇Permutations