Relative Frequency Form

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Relative Frequency Form

◇Overview

The Relative Frequency (Empirical / Frequentist) Form of Probability estimates the probability of an event $E$ by conducting $n$ independent, identical trials of a statistical experiment and computing the proportion of trials in which $E$ occurs. Under Bernoulli's Law of Large Numbers, this empirical ratio converges to the true probability $P(E)$ as $n \to \infty$ .

◇Key Definitions

Empirical Relative Frequency: If event $E$ occurs $f_n(E)$ times in $n$ trials, its relative frequency is:
$\hat{P}_n(E) = \frac{f_n(E)}{n}$
Frequentist Limit Definition: $P(E) = \lim_{n \to \infty} \frac{f_n(E)}{n}$ .
Consistency with Axioms: Relative frequencies automatically satisfy all three Kolmogorov axioms for any finite $n$ $n$ :
1. $0 \le f_n(E) \le n \implies 0 \le \hat{P}_n(E) \le 1$ .
2. $f_n(S) = n \implies \hat{P}_n(S) = \frac{n}{n} = 1$ .
3. If $A \cap B = \emptyset$ , then $f_n(A \cup B) = f_n(A) + f_n(B) \implies \hat{P}_n(A \cup B) = \hat{P}_n(A) + \hat{P}_n(B)$ .

◇Worked Example

Problem: A network engineer transmits $n = 2,500$ test packets across a transatlantic link and observes that $35$ packets are dropped ( $f_n(D) = 35$ ). Estimate the packet drop probability $\hat{P}(D)$ and successful delivery probability $\hat{P}(D^c)$ .

Solution:
$\hat{P}(D) = \frac{35}{2500} = 0.014 \quad (1.4\%), \qquad \hat{P}(D^c) = 1 - 0.014 = 0.986 \quad (98.6\%)$

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Next Topic: ◇Priori Probability
Related Notes: ◇Basic Formula of Probability, ◇Additive and Complementary Rules

CONFIRMATION // 理解度の確認

ACTIVE RECALL VERIFICATION

Select an answer to verify conceptual comprehension and reveal the technical rationale.

How is the relative frequency $\hat{P}_n(E)$ of an event $E$ computed after $n$ experimental trials?

What fundamental theorem guarantees that as the number of independent trials $n \to \infty$ , the empirical relative frequency $\frac{f_n(E)}{n}$ converges to the true theoretical probability $P(E)$ ?

Why is the Gambler's Fallacy (believing that after $5$ heads in a row on a fair coin, a tail is "due" on the $6$ th toss) a misunderstanding of the Law of Large Numbers?

In a Monte Carlo simulation of $n = 10,000$ random points in a unit square, $7,854$ points fall inside a quarter-circle. What is the relative frequency estimate of the probability of landing inside the quarter-circle?

For any finite number of trials $n \ge 1$ , if events $A$ and $B$ are mutually exclusive ( $A \cap B = \emptyset$ ), why does $\hat{P}_n(A \cup B) = \hat{P}_n(A) + \hat{P}_n(B)$ hold exactly?