Understanding Probability and Chance Behavior

Introduction to Probability: Getting it Wrong

Most of our understanding of probability comes from everyday games of chance and real-life familiarity (e.g., chance of rain). Children are often fascinated by simple chance outcomes in games like Candyland, but this intuition isn't enough for a statistical framework.

Statistical Probability vs. Daily Use

  • Statistical Definition: Probability studies what would happen if a situation were repeated many, many times.

  • Daily Definition: Often, we apply probability to a single event without considering repeated trials.

    • Example: If there's a 30\% chance of rain today, statistically this means if we were to relive today over and over, 30\% of those days would have rain. This contradicts our daily experience of living today only once.

Motivating Question: Tiny Tim's Piggy Bank

Scrooge asks Tiny Tim to get two coins from a piggy bank.

  • Scenario 1: Piggy bank with various coins (quarters, dimes, pennies, nickels)

    • Question: What kind of results can Tim get, and what is he likely to get?

    • Possible Outcomes: two quarters, two dimes, two pennies, a nickel and a penny, a quarter and a nickel, etc.

    • The set of all possible outcomes and their likelihoods is called the distribution of lunch money.

  • Scenario 2: Modified Piggy Bank (e.g., mostly pennies, one nickel)

    • The distribution of lunch money changes. Tim cannot get two dimes, and is almost certainly going to pull out two pennies.

    • The contents of the piggy bank represent the population of all coins.

    • When Tim shakes out two random coins, this is a Simple Random Sample (SRS) of size two.

    • Results like (two quarters), (a dime and a penny), or (two pennies) are three possible samples of size two.

    • Some samples (like two pennies in the modified bank) occur more frequently because of the population's composition.

  • The Real Goal: While we start by understanding what happens when we know the contents (population), the ultimate goal of statistics is to infer what is inside the piggy bank (population characteristics) by observing multiple samples (e.g., Tim shaking out two coins on Monday, Wednesday, and Friday).

Long-Run Behavior: The Coin Toss Experiment

Consider tossing a 'fair' coin many times:

  • Initial Variability: If you flip a coin 5 times, you might get 100\% heads or 20\% heads. The proportion of heads is highly variable at first. Each additional toss significantly changes the proportion.

  • Decreasing Variability: As you continue tossing the coin (hundreds, thousands of times), the weight of all previous results makes each new toss matter less. The proportion of heads stabilizes and gets closer to 50\%.

    • Analogy: Getting a zero on the first homework assignment significantly drops your class grade. Skipping a homework in week 14 has a much smaller impact on your final grade.

  • Definition of Probability: The proportion of tosses that land heads (e.g., 50\%) over a long series of trials is defined as the probability.

  • Key Insight: There is no pattern at all in the short run for individual flips, yet a predictable pattern emerges in the long run.

The Nature of Randomness

In statistics,