Home Class 11 Applied Maths Chapter 9: Probability Exercise 9.1: Random Experiments and Sample Space
Chapter 9 • Unit IV

Exercise 9.1: Random Experiments and Sample Space

Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).

9.1 Random Experiments, Sample Space, and Event Algebra

In Applied Mathematics, financial risk engineering, actuarial science, and quality control, decision-making occurs under conditions of uncertainty. Probability begins by establishing a rigorous set-theoretic sample space containing all conceivable outcomes of a random experiment.

Fundamental Definitions & Set-Theoretic Axioms

  • Random Experiment: An experiment is random if:
    1. It has more than one possible outcome.
    2. It is impossible to predict the exact outcome in advance with complete certainty.
  • Sample Space (\(S\)): The set of all possible elementary outcomes of a random experiment: \[ S = \{ \omega_1, \omega_2, \dots, \omega_n \} \]
  • Event (\(E\)): Any subset of the sample space \(S\): \[ E \subseteq S \]
  • Taxonomy of Events:
    • Impossible Event: The empty set \(\emptyset\) (cannot occur).
    • Sure (Certain) Event: The entire sample space \(S\) (guaranteed to occur).
    • Simple (Elementary) Event: A subset containing exactly one sample point \(\{\omega_i\}\).
    • Compound Event: A subset containing two or more distinct sample points.
  • Mutually Exclusive Events: Events \(A\) and \(B\) are mutually exclusive (disjoint) if their simultaneous occurrence is impossible: \[ A \cap B = \emptyset \]
  • Exhaustive Events: Events \(E_1, E_2, \dots, E_k\) are exhaustive if their union constitutes the entire sample space: \[ E_1 \cup E_2 \cup \dots \cup E_k = S \]

Standard Benchmark Sample Spaces in Applied Mathematics

Experiment Sample Space \(S\) Total Cardinality \(n(S)\)
Tossing \(n\) Coins \(\{ (x_1, x_2, \dots, x_n) : x_i \in \{H, T\} \}\) \(2^n\) (e.g. \(n=3 \implies 2^3 = 8\))
Throwing 2 Fair Dice \(\{ (i, j) : i, j \in \{1, 2, 3, 4, 5, 6\} \}\) \(6^2 = 36\) ordered pairs
Drawing from a Standard 52-Card Deck 4 Suits: Hearts, Diamonds (Red = 26); Clubs, Spades (Black = 26) 52 cards (12 Face cards: J, Q, K; 4 Aces)
Manufacturing Quality Inspection Testing 3 consecutive items for Defective (\(D\)) or Non-defective (\(N\)) \(2^3 = 8\) outcome sequences

Comprehensive Applied Example: Industrial Quality Control Sampling

Scenario: A quality assurance inspector examines 3 components selected at random from an assembly line. Each component is classified as either Defective (\(D\)) or Non-Defective (\(N\)).

  1. Write down the complete sample space \(S\).
  2. Define event \(A\): "Exactly one component is defective".
  3. Define event \(B\): "At least two components are defective".
  4. Define event \(C\): "No component is defective".
  5. Determine whether \(A\) and \(B\) are mutually exclusive. Are \(A, B, C\) exhaustive?

Solution:

  1. Sample Space: \[ S = \{ NNN, \, NND, \, NDN, \, DNN, \, NDD, \, DND, \, DDN, \, DDD \}, \quad n(S) = 8. \]
  2. Event \(A\) (Exactly 1 defective): \[ A = \{ NND, NDN, DNN \}, \quad n(A) = 3. \]
  3. Event \(B\) (At least 2 defective): \[ B = \{ NDD, DND, DDN, DDD \}, \quad n(B) = 4. \]
  4. Event \(C\) (No defective component): \[ C = \{ NNN \}, \quad n(C) = 1. \]
  5. Mutually Exclusive & Exhaustive Verification:
    • \(A \cap B = \emptyset\). Since they share no common sample points, \(A\) and \(B\) are mutually exclusive.
    • \(A \cup B \cup C = \{ NND, NDN, DNN, NDD, DND, DDN, DDD, NNN \} = S\).
    • Because their union equals \(S\) and their pairwise intersections are empty, the events \(A, B, C\) form a mutually exclusive and exhaustive partition of \(S\).
Chapter Hub Next: Ex 9.2