Sample Space and Probability

Definition

Where \( \mathbb{P}(A|B) \) is the probability of event \(A\) given that event \(B\) occurs:

$$ \mathbb{P}(A|B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} $$

Notice that if \(P(B) = 0\) then it’s meaningless to condition \(A\) on \(B\).

Satisfying the Axioms of Probability

Axiom 1: Normalization

$$ \mathbb{P}(B|B) = \frac{\mathbb{P}(B \cap B)}{\mathbb{P}(B)} = 1 $$

Axiom 2: Non-negativity

Because \(A \cap B \subseteq B \), then \( 0 \le \mathbb{P}(A|B) \le 1 \).

The Formula

By definition…

$$ \mathbb{P}(A) = \mathbb{P}(A \cap B) + \mathbb{P}(A \cap B^{c}) $$

From conditional probability …

$$ \mathbb{P}(A) = \mathbb{P}(A|B) \ \mathbb{P}(B) + \mathbb{P}(A|B^c) \ \mathbb{P}(B^c) $$

Therefore

$$ \mathbb{P}(B|A) = \frac{ \mathbb{P}(B \cap A) }{ \mathbb{P}(A) } $$ $$ = \frac{ \mathbb{P}(A|B) \mathbb{P}(B) }{ \mathbb{P}(A|B) \ \mathbb{P}(B) + \mathbb{P}(A|B^c) \ \mathbb{P}(B^c) } $$

Probability is a Fraction of a Finite Set

A simplified definition: a probability is a fraction of a finite set. For example, in the 2024 GSS survey, of the 3,986 respondents, 42 were bankers. If we choose a person from this population at random, the probability that they are a bank teller is 1.054%.