Probability Lecture

Jul 11, 2024

Probability Lecture Notes

Introduction to Probability

  • Probability (P(A)): The measure of the likelihood that an event A will occur.
  • Formula: Probability = (Number of favorable outcomes) / (Total possible outcomes)

Sample Space

  • Definition: Set of all possible outcomes in a probabilistic experiment.
  • Example (Flipping a Coin): Outcomes = Heads (H) or Tails (T).

Flipping Multiple Coins

  • Two Coins: Sample space = HH, HT, TH, TT
  • Tree Diagram: Visual representation to list all possible outcomes.
  • Three Coins:
    • Sample space: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
    • Total outcomes = 2^3 = 8

Probability Values Range

  • 0: Event will never occur.
  • 1: Event will always occur (100% chance).
  • Example: P(Event) = 0.3 means 30% chance of occurring.

Examples of Calculating Probability

Example 1: Flipping Coins

  1. Two Coins, P(at least one head):
    • Sample space: HH, HT, TH, TT
    • Favorable outcomes: HH, HT, TH
    • P(at least one head) = 3/4 = 0.75 (75% chance)
  2. Three Coins, P(at least two tails):
    • Sample space: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
    • Favorable outcomes: HTT, THT, TTH, TTT
    • P(at least two tails) = 4/8 = 0.5 (50% chance)
  3. Three Coins, P(exactly one tail):
    • Sample space: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
    • Favorable outcomes: HHT, HTH, THH
    • P(exactly one tail) = 3/8 = 0.375 (37.5% chance)

Example 2: Rolling a Six-Sided Die

  1. P(getting a 2):
    • Sample space: 1, 2, 3, 4, 5, 6
    • P(2) = 1/6 ≈ 0.167 (16.7% chance)
  2. P(getting a 3 or 5):
    • Favorable outcomes: 3, 5
    • P(3 or 5) = 2/6 = 1/3 ≈ 0.333 (33.3% chance)
  3. P(number ≤ 4):
    • Favorable outcomes: 1, 2, 3, 4
    • P(≤ 4) = 4/6 = 2/3 ≈ 0.667 (66.7% chance)
  4. P(number > 3):
    • Favorable outcomes: 4, 5, 6
    • P(> 3) = 3/6 = 1/2 = 0.5 (50% chance)
  5. P(number ≤ 5):
    • Favorable outcomes: 1, 2, 3, 4, 5
    • P(≤ 5) = 5/6 ≈ 0.833 (83.3% chance)

Conclusion

  • Probability Calculations: Straightforward process to determine the likelihood of events.
  • Further Learning: Check out additional resources on topics like independent/dependent events, mutually exclusive events, conditional probability, and more.