CONDITIONAL PROBABILITY

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

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◇Overview

Conditional Probability, denoted $P(A \mid B)$, measures the updated probability that event $A$ occurs given the partial knowledge that event $B$ has already occurred. Geometrically, conditioning on $B$ restricts the sample space from $S$ to $B$ and rescales all outcome probabilities within $B$ by $\frac{1}{P(B)}$.

◇Key Definitions

• Formal Definition: For any two events $A$ and $B$ with $P(B) > 0$:
  $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$
• Reduced Sample Space Interpretation: When outcomes in $S$ are equally likely, $P(A \mid B) = \frac{|A \cap B|}{|B|}$.
• Conditional Probability as a Measure: For fixed $B$ with $P(B) > 0$, $Q(A) = P(A \mid B)$ satisfies all three Kolmogorov axioms: $P(A \mid B) \ge 0$, $P(S \mid B) = 1$, and $P(A_1 \cup A_2 \mid B) = P(A_1 \mid B) + P(A_2 \mid B)$ for disjoint $A_1, A_2$.

◇Worked Example

Problem: A fair six-sided die is rolled. Given that the outcome is an even number ($B = \{2, 4, 6\}$), what is the conditional probability that the outcome is greater than 3 ($A = \{4, 5, 6\}$)?

Solution:

• Intersection: $A \cap B = \{4, 6\}$, so $P(A \cap B) = \frac{2}{6}$ and $P(B) = \frac{3}{6}$.
• $P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{2/6}{3/6} = \frac{2}{3}$.

◇References & Navigation

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• Related Notes: ◇Complementary Rule, ◇Multiplication Rule