Relative Frequency and Conditional Probability

calendar_today January 21, 2026 article 183 words access_time 1 min replay Modified February 10, 2026

Table of Contents

Relative Frequency Approach

$$f_{A}(n) = \frac{N_{n}(A)}{n}$$

Theorem 02.1 (Regularity of Statistics).

Given unlimited repetitions, the relative frequency of $A$ will approach the true probability of $A$.

$$P[A] = \lim_{ n \to \infty } f_{A}(n) = \lim_{ n \to \infty } \frac{N_{n}(A)}{n}$$

Reducing the sample space once you have information about an experiment’s outcome
Conditional probability of event $A$ given the occurrence of event $B$:
- Only defined for $P[B] > 0$
- $P[A \cap B]$ is the joint probability of $A$ and $B$
- $P[A|B]$ is the probability that $A$ occurs relative to a sample space of all the outcomes in $B$

$$P[A|B] = \frac{P[A \cap B]}{P[B]} = \frac{P[AB]}{P[B]}$$

Useful shortcut if $A$ is contained in $B$: Since $A \subset B \implies AB = A$, $P[A | B] = \frac{P[A]}{P[B]}$
- Example: if $A$ is the bottom of the deck being the ace of clubs and $B$ is the bottom of the deck being black, then $P[A|B] =\frac{P[A]}{P[B]}$ since all ace of clubs are black

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