Geog 2251 Lecture 5 W25 (1)

Probability Theory

  • Probability theory is the branch of mathematics that describes random behavior.

  • Originated from the study of games of chance.

  • Now widely used in various fields:

    • Stock markets

    • Human behavior

    • Population characteristics

    • Human mortality

    • Disease

    • Sports

Variability in Measurements

  • Natural variability exists in all measurements, making any computed statistic prone to differences from the true value.

  • Probability expresses this uncertainty.

  • E.g., a statistically significant result (95% confidence) implies less than a 5% chance that results are due to chance.

Random Phenomena

  • Individual outcomes are uncertain, yet outcomes display regular distributions over large repetitions.

  • Probability of a specific outcome = proportion of times that outcome occurs in extensive trials.

Definition of Probability

  • Probability measures the likelihood of an event occurring, calculated as:

    • Probability = Number of favorable cases / Total number of cases (Relative frequency)

  • Example: Probability of rolling a 6 on a 6-sided die = 1/6 = 0.17

Types of Probability

  1. Empirical Probability

    • Determined through repeated experimentation.

    • E.g., flipping a coin 500 times with heads appearing 262 times:(P(\text{heads}) = \frac{262}{500} = 0.524)

  2. Theoretical Probability

    • Based on possible outcomes and mathematical computations.

    • Example: Drawing a card from a standard deck where each card is equally likely.

  3. Subjective Probabilities

    • Adjusted based on personal experience and intuition, e.g., weather forecasts.

Law of Large Numbers

  • As sample size increases, empirical and theoretical probabilities converge, e.g., more coin tosses result in empirical probability approaching 0.5.

  • Distinct symbols for empirical and theoretical distributions (mean, standard deviation, variance).

Odds

  • Odds relate probability to occurrences and non-occurrences, mathematically expressed as ratios.

    • E.g., Odds of drawing a red marble from a bag of colored marbles.

Probability Rules

  1. Probability of an impossible event = 0.

  2. Probability of a certain event = 1.

  3. All probabilities lie between 0 and 1.

  4. The sum of probabilities of all possible outcomes = 1.

Addition and Multiplication Rules

  • Addition Rule: For mutually exclusive events, add individual probabilities.E.g., probability of getting heads or tails = 0.5 + 0.5 = 1.

  • Multiplication Rule: Probability that two events (A and B) both occur = P(A) x P(B).

  • Example: Probability of winning a lottery: (P = \frac{1}{49} \times \frac{1}{48} \times \frac{1}{47} \times \ldots)

Probability Distributions

  • Represented by histograms that show event frequencies and curve shapes:

    • Types of distributions include:

      • Continuous: Normal, T, Chi-Squared.

      • Discrete: Binomial, Poisson.

Continuous vs. Discrete Distributions

  • Continuous Variables: Take infinite values (e.g., height, temperature).

  • Discrete Variables: Take finite values (e.g., number of stores).

Binomial and Poisson Distributions

  • Binomial Distribution: Outcomes of a success/failure experiment (Bernoulli Trials).

    • e.g. Lottery wins.

  • Poisson Distribution: Addresses the probability of rare events per unit time or space.

    • e.g., number of earthquakes in a year.

The Normal Distribution

  • The most common distribution (bell curve), applies to many natural phenomena (e.g., heights, weights).

  • Characterized as:

    • Bell-shaped

    • Unimodal

    • Symmetrical

    • Mode, median, and mean are equal.

Empirical Rule of Normal Distribution

  • Approximately:

    • 68% within 1 standard deviation of the mean.

    • 95% within 2 standard deviations.

    • 99.7% within 3 standard deviations.

Standard Normal Distribution and Z-scores

  • Normal distributions can be standardized with Z-scores.

  • A Z-score indicates how many standard deviations an observation is from the mean.

  • E.g., IQ scores with mean 100 and std deviation 15.

    • 68% of scores lie within (85, 115).

Calculating Probabilities and Using Z-tables

  • After standardizing Z-scores, probabilities of x values are determined.

  • Probabilities are calculated from Z-tables, represent areas under the curve.

Interpreting Z-table Values

  • Area under the curve to the left of Z indicates the probability:

    • For example, P(Z=0) = 0.5, meaning half the area lies below the mean.

Example Calculations

  • Example calculation for heights in a normal distribution:

    • Mean height 163cm, if looking for the probability of a student being less than 170cm:

    • Find corresponding Z-value, use Z-table.

  • Results indicate probabilities for ranges:

    • Area left of 170 provides the probability of getting a height under 170cm.

Area Between Two Values

  • To find the probability between two heights (e.g., 165 and 170cm):

    • Area left of 170 – area left of 165 gives the probability between.

Height Probability Calculations

  • For height < 150cm or other specific values, Z-calculations correspond to exact probabilities using respective Z-tables.

    • E.g., Z = -2.17 relates to height calculations on the left.

    • Final computations will show overall probabilities in context and can include probability above/below mean.