Introduction to Probability

1. Basic Probability

Probability Scale

Probability is measured on a scale from $0$ to $1$:

  • $0$: Impossible event.
  • $1$: Certain event.
  • $0.5$: Equally likely as not.

Fundamental Rules

  • Complementary Events: The probability of an event NOT happening is $P(\text{not } A) = 1 - P(A)$.
  • Relative Frequency: An estimate of probability based on experimental data. $$\text{Relative Frequency} = \frac{\text{Number of successful trials}}{\text{Total number of trials}}$$
  • Expected Frequency: The number of times an event is expected to occur over a number of trials. $$\text{Expected Frequency} = P(A) \times \text{Total Trials}$$

2. Sample Space

A Sample Space is a list of all possible outcomes of an experiment.

  • Sample Space Diagrams: Tables used to list outcomes for two events (e.g., rolling two dice).
  • Calculation: $P(A) = \frac{\text{Number of outcomes satisfying } A}{\text{Total outcomes in sample space}}$.

Example: Sample space of rolling 2 dice Sample space table for two dice

3. Combined Events

Event Types

  • Independent Events: The outcome of one event does not affect the outcome of another.
  • Mutually Exclusive Events: Events that cannot happen at the same time.

Tools for Calculation

  • Venn Diagrams (2 Sets): Used to represent the relationship between two events.
    • Intersection ($\cap$): Both events occur.
    • Union ($\cup$): Either or both events occur. Venn Diagram
  • Tree Diagrams (With Replacement):
    • Each branch represents an outcome.
    • Multiply probabilities along the paths to find the probability of a combined sequence of events.
    • Sum the probabilities of the different paths that satisfy the required outcome.

Probability tree diagram with replacement