Probability and Events

Sample Space and Events

• Sample Space (S): The set of all possible outcomes of a random experiment.
• Event (A): A subset of the sample space.
• Event Space: The set of all possible events associated with a random experiment.

Random Experiment

• A process with a defined set of possible outcomes where the specific outcome is unpredictable.

Mutually Exclusive Events

• Two events that cannot occur simultaneously.
• Symbolically: A ∩ B = Φ (empty set) or P(A ∩ B) = 0

Classical Definition of Probability

• P(A): The probability of event A occurring.
• Formula: P(A) = m(A) / n(S), where:
  • m(A) is the number of outcomes favorable to A.
  • n(S) is the total number of possible outcomes in the sample space.

Axioms of Probability

• Axiom (i): P(A) ≥ 0 (Probability is non-negative)
• Axiom (ii): P(S) = 1 (Probability of the certain event is 1)
• Axiom (iii): For mutually exclusive events A1, A2, ... An, P(A1 ∪ A2 ∪ ... ∪ An) = P(A1) + P(A2) + ... + P(An)

Probability Notation

• P(A): Probability of event A.
• P(A̅) or P(Aᶜ): Probability of event A not occurring.
• P(A ∪ B) or P(A + B): Probability of at least one of the events A and B occurring.
• P(A ∩ B) or P(AB): Probability of both events A and B occurring.
• P(A | B): Conditional probability of event A given that event B has already occurred.

Properties of Probability

• 1. P(φ) = 0: The probability of the impossible event is 0.
• 2. P(A) ≤ 1: The probability of any event is less than or equal to 1.
• 3. P(A̅) = 1 − P(A): Probability of an event not occurring is 1 minus the probability of it occurring.
• 4. P(B ∩ A̅) = P(B) − P(B ∩ A): Useful to calculate probability of B occurring but A not occurring

Theorem of Addition

• For any events A and B: P(A + B) = P(A) + P(B) – P(AB).
• For mutually exclusive events A and B: P(AB) = 0, so P(A + B) = P(A) + P(B).
• For any three events A, B, and C:
  • P(A + B + C) = P(A) + P(B) + P(C) – P(BC) – P(CA) – P(AB) + P(ABC).
  • If mutually exclusive, P(A + B + C) = P(A) + P(B) + P(C).

Conditional Probability

• P(B | A): Probability of event B occurring given that event A has already occurred.
• Formula: P(B | A) = P(AB) / P(A), where P(A) > 0.

Theorem of Compound Probability

• P(AB): Probability of events A and B occurring simultaneously.
• Formula: P(AB) = P(A) * P(B | A).

Independent Events

• Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.
• Condition: P(AB) = P(A) * P(B).

Baye's Theorem

• Used to calculate the conditional probability of an event Ai given that another event X has occurred.
• Formula:
  • P(Ai | X) = [P(Ai) * P(X | Ai)] / [∑ P(Aj) * P(X | Aj)] (where j goes from 1 to n).
  • P(Ai) is the prior probability of Ai.
  • P(X | Ai) is the conditional probability of X given Ai.

Mathematical Expectation (E(x))

• The expected value of a random variable 'x' with n possible outcomes.
• Formula: E(x) = ∑ (pi * xi) (where 'i' denotes the index of the outcome, 'pi' is the probability of outcome 'i', and 'xi' is the value of 'x' for outcome 'i').

Variance (Var(x))

• Measures the spread of a random variable around its mean.
• Formula: Var(x) = E(x²) - [E(x)]²

Examples

• Example 1: Calculating the probability of a leap year having 53 Sundays.
• Example 2: Determining if two events are independent given their probabilities.
• Example 3: Applying Bayes' Theorem to calculate the probability of a certain event based on given conditions.
• Example 4: Determining the probability of at least one specific event occurring from two independent trials.

This quiz covers fundamental concepts in probability, including sample space, events, and mutually exclusive events. Understand classical definitions and axioms of probability through a variety of questions designed to test your knowledge and application skills.