Probability Concepts and Calculations

Probability

• An experiment is a repeatable process leading to various outcomes.
• An event is a collection of one or more outcomes, while the sample space (Ω) encompasses all possible outcomes of an experiment.
• Example of a sample space when rolling a die: Ω = {1, 2, 3, 4, 5, 6}.
• Events from the die roll:
  • E1 (even outcomes) = {2, 4, 6}
  • E2 (odd outcomes) = {1, 3, 5}
  • E3 (prime outcomes) = {2, 3, 5}.

Probability Scale

• The probability P(E) of an event E falls between 0 and 1: 0 ≤ P(E) ≤ 1.

Probability of an Event

• The probability P(E) of an event occurring is formulated as:
  • P(E) = (Number of ways E can occur) / (Total number of outcomes).
• Example: For a fair die, event A (rolling a multiple of 3: {3, 6}):
  • P(A) = 2 (favorable outcomes) / 6 (total outcomes) = 1/3.

Complementary Event

• For any event E, the complementary event E₀ signifies that E does not occur:
  • P(E₀) = 1 - P(E).

Probability Diagrams

• Visual aids for probability concepts include:
  • Sample space tables
  • Tree diagrams
  • Venn diagrams.

Example Probabilities

• When rolling two dice, the probability of obtaining a total of 8 is P(8) = 5/36.
• In a bag with 4 red, 5 yellow, and 11 blue balls, the probability of drawing a red and blue ball (in any order) is calculated using combinations.

Venn Diagrams

• Venn diagrams illustrate relationships between events:
  • A∪B indicates "A or B"
  • A∩B indicates "A and B".
• Addition Rule:
  • P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
  • Rearranged: P(A ∩ B) = P(A) + P(B) - P(A ∪ B).

Application Example

• In a class of 30 students with memberships in choir and band:
  • 7 in the choir, 5 in the band, 2 in both.
  • Probability of selecting a student not in the band: P(not in band) = 25/30 = 5/6.
  • Probability of selecting a student not in choir or band: P(not in either) = 20/30 = 2/3.

Vet Survey Example

• A vet surveyed 100 clients, results include:
  • 25 own dogs, 53 own cats, 40 own tropical fish.
  • Overlapping ownership includes:
    • 15 own dogs and cats,
    • 10 own cats and tropical fish,
    • 11 own dogs and tropical fish,
    • 7 own all three.