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.