Sample Space and Events

Sample Space

• The sample space \((Ω)\) is the set of all possible outcomes of a random experiment.
• An element \((ω ∈ Ω)\) represents a single outcome or realization.
• Examples of sample spaces:
  • All outcomes of flipping 5 coins
  • Number of heads in 5 coin flips
  • Daily high temperature
  • Number of emails received in an hour

Events and Subsets

• An event (A) is a subset of the sample space Ω.
• Example: For 3 coin flips:
  • \(Ω\) = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
  • Event A: First flip is heads = {HHH, HHT, HTH, HTT}
  • Event B: Second flip is tails = {HTH, HTT, TTH, TTT}

Set Operations in Probability

• Union \((A ∪ B)\): Outcomes in A or B.
  • More inclusive.
• Intersection \((A ∩ B)\): Outcomes in both A and B.
  • More restrictive.
• Complement \((Aᶜ)\): Outcomes not in A.

Venn Diagram Intuition

• Sample space \(Ω\) can be visualized as a rectangle.
• Events are subsets (circles) inside \(Ω\).
• Area of A / Area of \(Ω = P(A)\) when outcomes are equally likely.

Additional Derived Rules

• Complement Rule: \(P(Aᶜ) = 1 − P(A)\)
• Empty Set: \(P(∅) = 0\)
• Subset Rule: If \(A ⊆ B\), then \(P(A) ≤ P(B)\)
• Inclusion-Exclusion Principle: \(P(A ∪ B) = P(A) + P(B) − P(A ∩ B)\)
  • We subtract the intersection to avoid counting the overlapping area twice.
  • If the events are mutually exclusive this will be \(P(A ∩ B) = 0\).

Event Types

Certain Event

• \(P(S) = 1\)
• An event that always occurs.
• Denoted by the sample space SS.

Independent Events

• Occurrence of one event does not affect the other.
• Probability rule: \(P(A \cap B) = P(A) \cdot P(B)\)

Dependent Events

• Occurrence of one event does affect the probability of the other.
• Probability rule: \(P(A \cap B) \ne P(A) \cdot P(B)\)

Mutually Exclusive Events

• \(A \cap B = \emptyset\)
• Two or more events cannot occur at the same time.
• Example: Drawing a card that is either a heart or a club (not both).
• Probability rule: \(P(A \cup B) = P(A) + P(B)\)
• Mutually exclusive events are never independent (unless one has probability zero), because if \(A \cap B = 0\), then \(P(A \cap B) \ne P(A) \cdot P(B)\) (unless one of them is 0).
• Because mutually exclusive events cannot occur together, knowing that one event has happened means you know the other event definitely did not happen.

Mutually Inclusive Events

• \(A \cap B \ne \emptyset\)
• Two or more events can occur together (they may overlap).
• Example: Drawing a card that is a king or a heart (King of Hearts counts in both).

Impossible Event

• \(P(\emptyset) = 0\)
• An event that can never occur.
• Represented by the empty set \(\emptyset\).

Complementary Events

• \(A \cup A^c = S, A \cap A^c = \emptyset\)
• Two events \(A\) and \(A^c\), such that exactly one must occur.
• Probability rule: \(P(A^c) = 1 - P(A)\)