Sample Space, Tree Diagrams, and Probability Calculations

Sample Space Representation and Tree Diagrams

• Sample Space Definition:

  • The sample space represents the set of all possible outcomes resulting from a statistical experiment.
  • Examples of paired numerical outcomes in a multi-stage experiment include:
    • $(1, 6)$
    • $(6, 1)$
    • $(1, 2)$
    • $(6, 6)$

• Tree Diagram Visualization:

  • A tree diagram is a visual structure used to systematically list and determine all optimal outcomes across sequential stages of an experiment.
  • Branching pathways in a tree diagram allow for precise identification of specific outcomes that satisfy a given event condition.

Fundamental Counting Principle

• Multi-Step Experiments:
  • To find the total number of outcomes in a complete experiment sample space across multiple steps, multiply the total number of outcomes available at each individual step.
  • For a multi-step process where step 1 has $n_1$ outcomes, step 2 has $n_2$ outcomes, and step $k$ has $n_k$ outcomes, the total number of outcomes $N$ is given by:         $N = n_1 \times n_2 \times \dots \times n_k$

Probability Calculations and Empty Events

• Empty Events:

  • An event is defined as an empty event (or null set) when there are zero outcomes that satisfy or favor the specified condition.
  • When an event contains zero favorable outcomes, its probability is precisely zero:         $P(\emptyset) = 0$

• Calculating Probability:

  • The probability of a specific event occurring is calculated by taking the ratio of favorable outcomes to the total count of possible outcomes in the sample space:         $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes in sample space}}$