Probability Fundamentals: Events, Sample Space, and Rules

Experiment & Event

  • Experiment (trial): A process used to obtain an observation.

  • Event: An outcome of an experiment.

  • Simple Event: An outcome that cannot be simplified further.

Sample Space

  • Sample Space: The set of all possible simple events for an experiment.

  • Examples:

    • Flipping a coin twice: {HH,HT,TH,TT}\lbrace HH, HT, TH, TT \rbrace

    • Rolling a die once: {1,2,3,4,5,6}\lbrace 1, 2, 3, 4, 5, 6 \rbrace

Mutually Exclusive Events

  • Two events are mutually exclusive (disjoint) if they cannot occur together.

  • Example: "Elvis is alive" and "Elvis is dead."

Independent Events

  • Two events are independent if the occurrence of one has no effect on the occurrence of the other.

  • Example: "I had a doughnut for breakfast" and "Labor Day is a holiday."

  • Mutually exclusive events with \text{probability} > 0 cannot also be independent.

Addition Rule

  • For any two events A and B: P(A or B)=P(A)+P(B)−P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

  • If A and B are mutually exclusive: P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)

Multiplication Rule

  • If two events, A and B, are independent: P(A and B)=P(A)P(B)P(A \text{ and } B) = P(A)P(B)

  • Extends to any number of independent events.

  • If events are dependent: The probability of A and B happening together cannot be found by simply multiplying their individual probabilities.

Dependent vs. Independent Draws

  • Independent Draws: Occur when an item is replaced after being drawn (e.g., drawing a ball, putting it back, then drawing another).

  • Dependent Draws: Occur when an item is not replaced after being drawn, changing the probabilities for subsequent draws.

Birthday Probability

  • The probability that there is a common birthday in a group of size nn is: P(common birthday)=1−P(all have different birthdays)P(\text{common birthday}) = 1 - P(\text{all have different birthdays})

  • For nn people: P(all have different birthdays)=(365365)(364365)…(365−n+1365)P(\text{all have different birthdays}) = \left(\frac{365}{365}\right) \left(\frac{364}{365}\right) \ldots \left(\frac{365-n+1}{365}\right)

Conditional Probability

  • Definition: The probability of event A occurring given that event B has already occurred.

  • Notation: P(A∣B)P(A|B)

  • Formula: P(A∣B)=P(A and B)P(B)P(A|B) = \frac{P(A \text{ and } B)}{P(B)} (assuming P(B) > 0)

  • Relationship to Independence: If A and B are independent, P(A∣B)=P(A)P(A|B) = P(A).

  • Relationship to Mutually Exclusive Events: If A and B are mutually exclusive, P(A∣B)=0P(A|B) = 0.