Notes on Probability and Normal Distribution

Chapter 7: Probability and the Normal Distribution

Distributions
  • Empirical Distribution:
    • Scores that come from actual observations.
  • Theoretical Distribution:
    • Hypothesized scores based on mathematical models and logic.
Probability
  • Probability measures the likelihood of an event occurring, ranging from:
    • No possibility ($0.00$)
    • Certain event ($1.00$)
  • To calculate probability, find:
    • The number of ways the desired outcome can occur.
    • The total possible outcomes.
Basic Probability Example: Coin Flip
  • Outcomes:
    • Heads, Tails (2 possible outcomes)
  • Probability of Heads:
    • P(Heads)=12=0.5P(Heads) = \frac{1}{2} = 0.5
Theoretical Distribution of a Deck of Cards
  • Frequency of each card (Ace-10, J, Q, K) is equal:
    • Probability of drawing any one card: 113\frac{1}{13}
  • Frequency distribution is rectangular with values around 0.0770.077 for each card.
Empirical Distribution of Card Draws
  • Based on drawing 52 cards from a deck:
    • Each card's frequency represented.
Binomial Distribution
  • Defined as the frequency distribution of events with only two outcomes:
    • Example: Tossing coins (Heads or Tails)
  • For multiple coin tosses:
    • For 3 coins, probability of all heads:
    • P(3Heads)=(12)3=18=0.125P(3 Heads) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} = 0.125
Theoretical Binomial Distribution Example
  • Results from tossing 3 coins:
    • Frequency of obtaining heads:
    • 0 heads: 0.125
    • 1 head: 0.375
    • 2 heads: 0.375
    • 3 heads: 0.125
Probability Calculation with Marbles
  • Example jar contents:
    • 3 Red, 6 Black, 5 Blue, 2 Yellow, 4 Green marbles (total = 20 marbles)
  • Probability of drawing:
    • Blue marble: P(Blue)=520=0.25P(Blue) = \frac{5}{20} = 0.25
    • Black or Green marble: P(Black∪Green)=1020=0.5P(Black \cup Green) = \frac{10}{20} = 0.5
    • Red, Yellow, Black, or Green: P(Red∪Yellow∪Black∪Green)=1520=0.75P(Red \cup Yellow \cup Black \cup Green) = \frac{15}{20} = 0.75
    • Alternatively, 1−P(NoBlue)=1−0.25=0.751 - P(No Blue) = 1 - 0.25 = 0.75
Normal Distribution
  • Characterized as:
    • Bell-shaped curve.
    • Theoretical distribution predicting the frequency of events.
Z-Scores
  • A Z-score represents:
    • A score expressed in standard deviation units.
  • Now applied at a population level rather than just a sample level.
Normal Distribution of Z-Scores
  • Graphs showing Z-scores from -3 to 3 fit into the shape of a normal distribution curve:
    • Area under the curve represents total probability.
Area Under the Curve
  • The total area = 100% (or $1.00$) indicates that all scores fall beneath the normal curve.
  • Z-scores allow calculation of an individual score’s probability in the context of this normal distribution, using pre-calculated tables for reference.