Probability

Learning Objectives

• Understand the concept of Sample Space and Events as the universal set for an experiment and define events as subsets of this space, including impossible, sure, simple, and compound events.
• Apply Algebra of Events using set operations like union, intersection, and complement, and comprehend mutually exclusive and exhaustive events.
• Grasp the Axiomatic Probability approach, defining probability as a function satisfying non-negativity, normalization, and additivity for mutually exclusive events.
• Calculate the Probability of Events using the axiomatic approach, including probabilities of 'A or B', 'not A', and equally likely outcomes.
• Analyze Mutually Exclusive and Exhaustive Events to understand events that cannot occur simultaneously and sets of events covering all possible outcomes.
• Utilize formulas for Probability of Union and Complement of Events, such as $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ and $P(A') = 1 - P(A)$, with Venn-diagram interpretation.
• Determine probabilities in Equally Likely Outcomes and Classical Probability scenarios using $P(E) = \frac{n(E)}{n(S)}$ for dice, coins, cards, and other selections.
• Validate Assignment of Probabilities ensuring $0 \leq P(\omega_i) \leq 1$ and total probability equals 1.
• Solve Combinatorics-Based Probability Problems using arrangements and combinations for scenarios like committees, lotteries, and card hands.
• Apply the formula for Probability of Three Events such as $P(A \cup B \cup C)$ and related event-union problems.

Chapter Notes

Sample Space and Events

• Sample Space: The universal set for an experiment.
• Events: Subsets of the sample space.
  • Impossible Event: The empty set $\Phi$.
  • Sure Event: The whole sample space $S$.
  • Simple Event: Contains only one sample point.
  • Compound Event: Contains more than one sample point.

Algebra of Events

• Complementary Event: For event $A$, the complement is $A' = S - A$.
• Union of Events: $A \cup B$ includes all outcomes in $A$ or $B$ or both.
• Intersection of Events: $A \cap B$ includes outcomes common to both $A$ and $B$.
• Difference of Events: $A - B = A \cap B'$ includes outcomes in $A$ but not in $B$.

Axiomatic Probability

• Axioms:
  1. $P(E) \geq 0$ for any event $E$.
  2. $P(S) = 1$.
  3. For mutually exclusive events $E$ and $F$, $P(E \cup F) = P(E) + P(F)$.

Probability of Events

• Probability Calculation:
  • For any event $A$, $P(A) = \sum P(w_i)$ for all $w_i \in A$.
  • Equally Likely Outcomes: $P(E) = \frac{n(E)}{n(S)}$.

Mutually Exclusive and Exhaustive Events

• Mutually Exclusive: $A \cap B = \Phi$.
• Exhaustive Events: $E_1 \cup E_2 \cup \ldots \cup E_n = S$.

Probability of Union and Complement of Events

• Union Formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Complement Formula: $P(A') = 1 - P(A)$.

Equally Likely Outcomes and Classical Probability

• Classical Probability: $P(E) = \frac{n(E)}{n(S)}$ when all outcomes are equally likely.

Valid Assignment of Probabilities

• Conditions:
  • $0 \leq P(w_i) \leq 1$ for each $w_i \in S$.
  • $\sum P(w_i) = 1$.

Combinatorics-Based Probability Problems

• Use of Combinations and Permutations: For problems involving arrangements and selections.

Probability of Three Events

• Formula: $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)$.