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MATH 153 - STATISTICAL METHODS

LECTURE FOUR: PROBABILITY

INTRODUCTION
  • Instructor: Vincent K. Dedu
  • Department: Statistics and Actuarial Science
ABRAHAM DE MOIVRE
  • Lifespan: May 26, 1667 - November 27, 1754
  • Notable Contributions:
    • Known for de Moivre's formula, which links complex numbers to trigonometry.
    • Significant work in probability theory and normal distribution.
  • Interesting Fact: De Moivre predicted his own death based on his increasing sleep duration at night, calculating it would reach 24 hours total on November 27, 1754.
PROBABILITY OF AN EVENT
  • Definition:
    • The probability of an event A, denoted as P(A), measures the likelihood of occurrence of event A.
    • If P(A) = 0, Event A is impossible.
    • If P(A) = 1, Event A is certain.
    • If P(A) = 0.5, Event A is as likely to occur as not.
DEFINITIONS OF PROBABILITY
THREE MAIN SCHOOLS OF THOUGHT
  1. Classical Definition:
    • Prior probability; determined before any experiment is performed to observe the outcomes of event A.
  2. Empirical Definition:
    • Based on relative frequencies of past occurrences to forecast future probabilities, meaning P(A) is determined after observing event A.
    • Formula:
      P(A)=n(A)n(S)P(A) = \frac{n(A)}{n(S)}
    • Where n(A) = number of successful outcomes, n(S) = total number of possible outcomes.
  3. Subjective Definition:
    • Based on personal belief or opinion through available evidence.
TERMINOLOGIES
  • Key terms include:
    • Experiment
    • Outcomes
    • Trial
    • Sample space
    • Event
EXAMPLE PROBLEM
  • Problem: A couple planning to have three children.
    • Task: List all possible outcomes.
    • Probability of exactly two girls?
    • Possible Outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG
    • Tree Diagram will visually depict outcomes.
PROBABILITY OF COMPOUND EVENTS
  • Definition of Events:
    • Two or more events can be combined using set operations ∩ (intersection) and ∪ (union).
    • Union: A ∪ B occurs if A, B, or both occur.
    • Intersection: A ∩ B occurs if both A and B occur simultaneously.
DEFINITIONS OF COMPOUND EVENTS
  1. Mutually Exclusive Events:
    • Events that cannot occur at the same time; no common outcomes (P(A ∩ B) = 0).
    • If A and B are mutually exclusive:
      P(A∪B)=P(A)+P(B)P(A ∪ B) = P(A) + P(B)
  2. Exhaustive Events:
    • Events that partition the sample space S; covers all possible outcomes.
  3. Independent Events:
    • Events where the occurrence of one does not affect the occurrence of the other.
    • Mathematically:
      P(A∩B)=P(A)imesP(B)P(A ∩ B) = P(A) imes P(B)
    • Dependent events:
      P(A∩B)=P(A)imesP(B∣A)P(A ∩ B) = P(A) imes P(B|A)
  4. Conditional Probability:
    • The probability of A given B has occurs, denoted P(A|B).
    • Defined by:
      P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A ∩ B)}{P(B)}
ADDITION RULE
  • For events A and B:
    • If both events’ probabilities are given (P(A) and P(B)), then:
      P(A∪B)=P(A)+P(B)−P(A∩B)P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
EXAMPLES
Example 1:
  • Events A and B with probabilities:
    • P(A) = 1930\frac{19}{30}
    • P(B) = 25\frac{2}{5}
    • P(A ∩ B) = 45\frac{4}{5}
    • Find P(A ∪ B):
      P(A∪B)=P(A)+P(B)−P(A∩B)P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
      P(A∪B)=1930+25−45=730P(A ∪ B) = \frac{19}{30} + \frac{2}{5} - \frac{4}{5} = \frac{7}{30}
Example 2:
  • In a group of 20 adults, find:
    • P(W), probability of selecting a woman.
    • Let W = event of selecting a woman, G = event wearing glasses.
    • Use addition rule:
      P(W∪G)=P(W)+P(G)−P(W∩G)P(W ∪ G) = P(W) + P(G) - P(W ∩ G)
MUTUALLY EXCLUSIVE EVENTS
  • Formula:
    • For mutually exclusive events, where P(A ∩ B) = 0:
      P(A∪B)=P(A)+P(B)P(A ∪ B) = P(A) + P(B)
COMPLEMENTS
  • Defined as the event where A does not occur:
    • A′=1−P(A)A' = 1 - P(A)
    • Example: For a die, if A = {1, 2}, then S = {1, 2, 3, 4, 5, 6}, and therefore P(A ∪ B) = 1.
EXAMPLES OF COMPLEMENT
  • Drawing a card from a pack of 52: Find probability for A (is seven) and A' (is not seven).
  • Therefore:
    • Let P = probability of drawing seven, then P' = 1−P(A)=12131 - P(A) = \frac{12}{13}
EXAMPLE OF RAIN PREDICTION
  • If P(A) = probability it rains tomorrow = 0.20, then probability it does not rain:
    • P′(A)=1−P(A)=1−0.2=0.8P'(A) = 1 - P(A) = 1 - 0.2 = 0.8
APPLICATIONS OF PROBABILITY THEORY
  • Industries use probability for reliability assessments.
  • Government predictions regarding fiscal policies.
  • Theoretical physics applications in quantum mechanics.
  • Genetics research by biologists.
  • Insurance companies for premiums and policy predictions.
  • Business and manufacturing decisions based on probability estimations.
  • Investors use probability to assess stock growth potential.
CONCLUSION
  • Final insights into the pervasive role of probability theory across multiple fields and its implications for decision making.
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