Class 11 Mathematics: Probability

• Introduction to Probability

  • Definition: Measure of uncertainty of an event
  • Historical Context: Based on games of chance
  • Approach: Axiomatic Approach (modern standard)

• Random Experiments and Outcomes

  • Random Experiment: Outcome cannot be predicted with certainty
  • Outcome: Possible result of a random experiment
  • Sample Space ($S$): Set of all possible outcomes
    • Example (Coin toss): $S = {H, T}$
    • Example (Die roll): $S = {1, 2, 3, 4, 5, 6}$
  • Event ($E$): A subset of the sample space ($E \subset S$)
    • Elementary Event: Contains only a single outcome
    • Compound Event: Contains more than one outcome

• Types of Events

  • Impossible Event: Null set ($\emptyset$), probability is $0$
  • Sure Event: Entire sample space ($S$), probability is $1$
  • Complementary Event: "Not $E$" ($E'$ or $\bar{E}$)
  • Mutually Exclusive Events: Disjoint events ($A \cap B = \emptyset$)
  • Exhaustive Events: Union of events equals sample space ($\bigcup E_i = S$)

• Axiomatic Approach to Probability

  • Let $S = {\omega_1, \omega_2, \dots, \omega_n}$
  • Conditions for Probability Function $P(\omega_i)$:
    • $0 \le P(\omega_i) \le 1$ for each $\omega_i$
    • $\sum P(\omega_i) = P(S) = 1$
    • For any event $A$, $P(A) = \sum P(\omega_i)$ for all $\omega_i \in A$
  • Equally Likely Outcomes: $P(\omega_i) = \frac{1}{n}$

• Probabilities of Compound Events

  • Addition Rule (General): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
  • Mutually Exclusive Case: If $A \cap B = \emptyset$, then $P(A \cup B) = P(A) + P(B)$
  • Complementary Rule: $P(\bar{A}) = 1 - P(A)$

• Important MP Board Exam Formulas

  • $P(A \text{ or } B) = P(A \cup B)$
  • $P(A \text{ and } B) = P(A \cap B)$
  • $P(\text{not } A) = P(\bar{A}) = 1 - P(A)$
  • $P(A \text{ but not } B) = P(A - B) = P(A) - P(A \cap B)$