Day 5

Math 216: Statistical Thinking

Bastola

Introduction to Probability

  • Experiment: Any act of observation with uncertain outcomes (e.g., tossing a coin).
  • Observation: The result of an experiment (e.g., heads or tails).
  • Probability: A measure of how likely an event is to occur, often thought of as “chance” or “likelihood.”
    • Example: The chance of rain tomorrow, the likelihood of rolling a 6 on a die.

Mathematical Representation

  • Let \(E\) represent an experiment.
  • An outcome of \(E\) is an element of the sample space \(S\).
  • Probability of an event \(A\) is denoted by \(P(A)\).

Key Definitions

Sample Point

  • Definition: The most basic outcome of an experiment.
  • Example: In a coin toss, each toss that results in H (Head) or T (Tail).


Sample Space

  • Definition: Collection of all possible outcomes (sample points) of an experiment.
  • Example: In a coin toss, the sample space is \(S = \{H, T\}\).


Event

  • Definition: A collection of sample points.
  • Example: Rolling an even number on a die.

Visualizing Probability

Venn Diagram

  • Sample Space (S): Represented as a closed figure in the diagram.
  • Sample Points: Each possible outcome shown as a solid dot within the figure.
  • Example: Coin Toss
    • Venn Diagram: Circle labeled S containing two points labeled H (Head) and T (Tail).
    • Representation: Helps visualize all possible outcomes in a single view.

Tree Diagram

  • Example: Tree diagram of a coin toss.

Tree diagram of a coin toss

Experiments and Sample Spaces

Experiment: Tossing a Coin

  • Sample Space (S): \[S = \{H, T\}\]

Experiment: Rolling a Die

  • Sample Space (S): \[S = \{1, 2, 3, 4, 5, 6\}\]

Experiment: Tossing Two Coins

  • Sample Space (S): \[S = \{HH, HT, TH, TT\}\]

Probability Rules and Calculations

Probability of a Sample Point

  • A number between 0 and 1 indicating the likelihood of the outcome.

Rules of Probability

  1. Range of Probabilities:

    • Each probability \(p_i\) must satisfy: \[0 \leq p_i \leq 1\]

  2. Sum of Probabilities:

    • The total probability across all sample points must sum to 1: \[\sum_{\text{all } i} p_i = 1\]

Steps for Calculating Event Probabilities

  1. Define the Experiment: Describe the observation process.
  2. List Sample Points: Identify possible outcomes.
  3. Assign Probabilities: Allocate probability to each sample point.
  4. Identify Event’s Sample Points: Determine which sample points are part of the event.
  5. Calculate Event Probability: Sum the probabilities of the event’s sample points.

Probability in a Dice Game

  • Experiment: Tossing a fair die.
  • Winning Condition: If the up face is an even number (2, 4, 6), you win $1.
  • Losing Condition: If the up face is odd (1, 3, 5), you lose $1.

Sample Space and Probabilities

  • Sample Space (S): \[S = \{1, 2, 3, 4, 5, 6\}\]
  • Uniform Probabilities: \[p_1 = p_2 = p_3 = \cdots = p_6 = \frac{1}{6}\]

Calculation of Winning Probability

  • Event of Winning:

    \[\text{Winning} = \{2, 4, 6\}\]

  • Probability of Winning:

    \[\text{Prob of Winning} = p_2 + p_4 + p_6 = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\]

Compound Events

Unions

  • Definition: Union of two events \(A\) and \(B\) (\(A \cup B\))
  • Occurs If: Either event \(A\), event \(B\), or both happen.
  • Composition: Includes all sample points in \(A\), \(B\), or both. \[A \cup B = \{x : x \in A \text{ or } x \in B\}\]


Intersections

  • Definition: Intersection of two events \(A\) and \(B\) (\(A \cap B\))
  • Occurs If: Both event \(A\) and event \(B\) happen.
  • Composition: Includes all sample points that are in both \(A\) and \(B\). \[A \cap B = \{x : x \in A \text{ and } x \in B\}\]

Example: Calculating Unions and Intersections in a Die-Toss

  • Event A: Toss an even number. \[A = \{2, 4, 6\}\]
  • Event B: Toss a number less than or equal to 3. \[B = \{1, 2, 3\}\]
  • Sample Space (S): \[S = \{1, 2, 3, 4, 5, 6\}\]
  • Union (\(A \cup B\)): \[A \cup B = \{1, 2, 3, 4, 6\}\]
  • Intersection (\(A \cap B\)): \[A \cap B = \{2\}\]
  • Probability of Union (\(P(A \cup B)\)): \[P(A \cup B) = \frac{5}{6}\]
  • Probability of Intersection (\(P(A \cap B)\)): \[P(A \cap B) = \frac{1}{6}\]