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Probability: Structured Notes

1. Introduction to Probability

  • Definition: Probability is the measure of the likelihood or chance of an event happening. It quantifies uncertainty and helps in predicting outcomes.

  • Range: Probability of an event is a number between 0 and 1.

    • 0 means the event will not happen.

    • 1 means the event will certainly happen.

    • 0.5 represents an event that is equally likely to occur or not occur.

2. Basic Terms in Probability

  • Experiment: A process that results in one or several outcomes.

    • Example: Rolling a die.

  • Sample Space (S): The set of all possible outcomes of an experiment.

    • Example: For a die roll, S = {1, 2, 3, 4, 5, 6}.

  • Event (E): A specific outcome or a group of outcomes of an experiment.

    • Example: Rolling an even number on a die, E = {2, 4, 6}.

  • Outcome: A single result of an experiment.

    • Example: Rolling a 3 on a die.

3. Types of Events

  • Simple Event: An event that consists of only one outcome.

    • Example: Rolling a 5 on a die.

  • Compound Event: An event that consists of more than one outcome.

    • Example: Rolling an even number on a die.

  • Complementary Event: The event that the event does not occur.

    • Formula: P(E') = 1 - P(E).

    • Example: The complement of rolling a 2 on a die is rolling any number except 2, {1, 3, 4, 5, 6}.

  • Mutually Exclusive Events: Two events that cannot happen at the same time.

    • Example: Tossing a coin cannot result in both heads and tails.

  • Independent Events: Two events are independent if the outcome of one event does not affect the outcome of the other.

    • Example: Tossing a coin and rolling a die.

  • Dependent Events: Two events are dependent if the outcome of one event affects the outcome of the other.

    • Example: Drawing two cards from a deck without replacement.

4. Probability Formula

  • Formula: The probability of an event P(E) is defined as:
    P(E) = (Number of favorable outcomes) / (Total number of possible outcomes)

5. Probability Rules

  • Addition Rule (for Mutually Exclusive Events):
    P(A ∪ B) = P(A) + P(B)

  • General Addition Rule (for any events):
    P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

  • Multiplication Rule (for Independent Events):
    P(A ∩ B) = P(A) × P(B)

  • Conditional Probability: The probability of event A occurring, given that event B has occurred:
    P(A | B) = P(A ∩ B) / P(B)

  • Complementary Rule:
    P(A') = 1 - P(A)

6. Special Distributions in Probability

  • Uniform Distribution: Every outcome in the sample space has the same probability of occurring.

    • Example: Rolling a fair die or tossing a fair coin.

  • Binomial Distribution: Describes the number of successes in a fixed number of independent trials of a binary (success/failure) experiment.

    • Formula:
      P(X = k) = (n choose k) × p^k × (1 - p)^(n - k)

    • Where:

      • n = number of trials

      • k = number of successes

      • p = probability of success on a single trial

  • Normal Distribution: A continuous probability distribution that is symmetrical around the mean, often referred to as the bell curve.

    • Defined by the mean μ and the standard deviation σ.

7. Venn Diagrams and Probability

  • Venn Diagrams: Used to visually represent sets and their relationships, especially when dealing with union and intersection of events.

    • Union (A ∪ B): Represents the set of outcomes that belong to either event A, event B, or both.

    • Intersection (A ∩ B): Represents the set of outcomes that are common to both event A and event B.

8. Expected Value

  • Expected Value (Mean): The average outcome of an experiment over many trials.

    • Formula (for discrete random variables):
      E(X) = Σ [x × P(x)]

    • Where x represents a possible outcome, and P(x) is the probability of that outcome.

9. Law of Total Probability

  • Law of Total Probability: Used to calculate the total probability of an event by considering all possible ways the event can happen, partitioning the sample space into mutually exclusive events.
    P(A) = Σ P(A | Bᵢ) × P(Bᵢ)

    • Where {Bᵢ} forms a partition of the sample space.

10. Examples of Probability Calculations

  • Example 1: Tossing a coin

    • Sample space: S = {H, T} (H = heads, T = tails).

    • Probability of heads: P(H) = 1/2, Probability of tails: P(T) = 1/2.

  • Example 2: Rolling a fair die

    • Sample space: S = {1, 2, 3, 4, 5, 6}.

    • Probability of rolling an even number: P(E) = 3/6 = 1/2.

  • Example 3: Drawing a card from a standard deck of 52 cards

    • Probability of drawing a heart: P(heart) = 13/52 = 1/4.

Example 4: Flipping a coin
Probability of getting heads: P(heads) = 1/2.