Probability, Sampling Variability, and the Law of Large Numbers

Unit Context: Unit 2 focuses on probability, its underlying principles, and its theoretical and practical applications.
Definition of Probability: Probability represents the long-run relative frequency of an outcome over a very large number of repetitions.
Relative Frequency Formula: $\text{Relative Frequency} = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Trials}}$
Application to Free-Throw Performance: In a free-throw shooting scenario, relative frequency is calculated as: $\text{Relative Frequency} = \frac{\text{Makes}}{\text{Total Shots}}$

Scenario Baseline: Consider a basketball player characterized as a $70\%$ ( $0.70$ ) free-throw shooter.
Interpretation of a $70\%$ Shooter:
- Each individual free throw shot taken has an independent $70\%$ probability ( $0.70$ chance) of going in.
Expected Outcomes vs. Guarantees:
- Out of $100$ attempted shots, the theoretical expected number of made shots is: $\text{Expected Makes} = 100 \times 0.70 = 70$
- Making exactly $70$ shots out of $100$ attempts is not guaranteed. Natural random variation causes empirical results to deviate from exact theoretical values.
Occurrence of Streaks:
- Common Misconception: Observers frequently assume that a shooter will not experience a continuous streak of success, such as making $10$ shots in a row.
- Probabilistic Reality: Streaks are actually very common in random processes. Consecutive successes occur naturally within probability distributions without requiring any change in underlying skill level.

Short-Run Characteristics:
- Unpredictability: Outcomes in the short run (a small number of repetitions or trials) are inherently unpredictable.
- High Variability: Short-run data exhibits extreme variability, deviating significantly from the theoretical percentage in multiple directions.
- Sample Size Constraint: In the short run, there are insufficient repetitions to stabilize the observed proportion of outcomes.
Long-Run Characteristics:
- Predictability: As the number of trials increases significantly, the overall proportion of outcomes stabilizes and becomes highly predictable.
- Convergence: Over a long sequence of repetitions, the observed cumulative relative frequency approaches the true underlying probability of $70\%$ ( $0.70$ ).
Graphical Behavior of Cumulative Proportion:
- Graphs tracking success proportions show large fluctuations during early shots due to short-run unpredictability.
- As total shots increase toward $100$ or more, cumulative success rates dampen their fluctuations and stabilize around the true average line of $70\%$ ( $0.70$ ).

Simulation Mechanics:
- Simulations set at a baseline success parameter of $70\%$ permit testing individual shots or batches of shots (e.g., $50$ shots at a time) up to $100$ total shots or more.
Empirical Demonstrations of Variability:
- Trial 1: A simulation run of $100$ shots may fail to yield exactly $70$ made shots due to short-term probabilistic variation.
- Trial 2: Re-running the simulation under identical parameters ( $70\%$ baseline over $100$ shots) resulted in exactly $74$ made shots, producing a distinct graphical curve with multiple sustained success streaks.
Sampling Variability:
- Defined as the natural variability observed between different random samples taken from the same underlying process or population.
- Every unique random sample of $100$ shots will produce different outcome totals and unique graphical patterns.

Probability Range Limits:
- All probability values $P$ must fall within the range of $0$ to $1$ inclusive: $0 \le P \le 1$
- Expressed as percentages, probabilities strictly range from $0\%$ to $100\%$ inclusive: $0\% \le P \le 100\%$
- An event with a probability of $0$ ( $0\%$ ) is impossible, while an event with a probability of $1$ ( $100\%$ ) is guaranteed to occur.
Short Run vs. Long Run Summary:
- Short Run: Unpredictable due to small sample size; high variation.
- Long Run: Predictable due to large sample size; outcomes stabilize near theoretical expectation.
Law of Large Numbers (LLN):
- Formal Definition: The Law of Large Numbers states that as the number of independent trials ( $n$ ) increases, simulated probabilities (relative frequencies) tend to get closer and closer to the true underlying percentage or theoretical average.