Class 11 Mathematics Notes Chapter 16 (Chapter 16) – Examplar Problems (English) Book

Detailed Notes with MCQs of Chapter 16: Probability from your NCERT Exemplar. This is a crucial chapter, not just for your Class 11 exams, but also forms the foundation for many topics in competitive government exams. Pay close attention to the definitions and the axiomatic approach.

Chapter 16: Probability - Detailed Notes for Government Exam Preparation

1. Basic Concepts

• Random Experiment: An experiment whose outcome cannot be predicted with certainty, but all possible outcomes are known.
  • Example: Tossing a fair coin, rolling an unbiased die, drawing a card from a well-shuffled deck.
• Outcome: A possible result of a random experiment.
  • Example: Getting 'Heads' when tossing a coin; getting a '4' when rolling a die.
• Sample Space (S): The set of all possible outcomes of a random experiment. It is denoted by 'S'.
  • Example:
    • Tossing a coin: S = {H, T}
    • Rolling a die: S = {1, 2, 3, 4, 5, 6}
    • Tossing two coins: S = {HH, HT, TH, TT}
  • The number of elements in the sample space is denoted by n(S).

2. Events

• Event (E): Any subset of a sample space S. Events are usually denoted by capital letters like A, B, E, etc.
  • Example: Getting an even number when rolling a die. E = {2, 4, 6}, which is a subset of S = {1, 2, 3, 4, 5, 6}.
• Elementary Event: An event consisting of only one outcome (a singleton subset of S).
  • Example: Getting exactly '3' when rolling a die. {3} is an elementary event.
• Compound Event: An event consisting of more than one outcome.
  • Example: Getting an odd number when rolling a die. E = {1, 3, 5}.
• Impossible Event (∅ or {}): An event that cannot occur. It is represented by the empty set. P(∅) = 0.
  • Example: Getting a '7' when rolling a standard die.
• Sure Event (S): An event that is certain to occur. It is represented by the entire sample space S. P(S) = 1.
  • Example: Getting a number less than 7 when rolling a standard die.

3. Algebra of Events

Let A and B be two events associated with a sample space S.

• Complementary Event ('not A' or A' or Aᶜ): The set of all outcomes in S that are not in A. A' = S - A.
• Event 'A or B' (A ∪ B): The set of all outcomes which are in A, or in B, or in both. Corresponds to the occurrence of at least one of the events A or B.
• Event 'A and B' (A ∩ B): The set of all outcomes which are common to both A and B. Corresponds to the simultaneous occurrence of events A and B.
• Event 'A but not B' (A - B): The set of all outcomes which are in A but not in B. A - B = A ∩ B'.
• Mutually Exclusive Events: Two events A and B are mutually exclusive if the occurrence of one prevents the occurrence of the other. They cannot happen simultaneously. Mathematically, A ∩ B = ∅.
  • Example: When rolling a die, the event of getting an even number {2, 4, 6} and the event of getting an odd number {1, 3, 5} are mutually exclusive.
  • Important Note: Elementary events are always mutually exclusive.
• Exhaustive Events: A set of events E₁, E₂, ..., E<0xE2><0x82><0x99> is said to be exhaustive if their union is the entire sample space S. That is, E₁ ∪ E₂ ∪ ... ∪ E<0xE2><0x82><0x99> = S. This means at least one of these events must occur.
  • Example: When rolling a die, the events A = {1, 2, 3} and B = {3, 4, 5, 6} are exhaustive because A ∪ B = {1, 2, 3, 4, 5, 6} = S. (Note: They are not mutually exclusive as '3' is common).
  • Example: Events {1}, {2}, {3}, {4}, {5}, {6} are mutually exclusive and exhaustive.

4. Axiomatic Approach to Probability

Let S be the sample space and P be a probability function defined on the subsets (events) of S. P satisfies the following axioms:

• Axiom 1 (Non-negativity): For any event E, P(E) ≥ 0.
• Axiom 2 (Certainty): P(S) = 1.
• Axiom 3 (Additivity for Mutually Exclusive Events): If E₁ , E₂, ..., E<0xE2><0x82><0x99> are mutually exclusive events (i.e., Eᵢ ∩ E<0xE2><0x82><0x8D> = ∅ for i ≠ j), then P(E₁ ∪ E₂ ∪ ... ∪ E<0xE2><0x82><0x99>) = P(E₁) + P(E₂) + ... + P(E<0xE2><0x82><0x99>).

5. Important Results Derived from Axioms

• P(∅) = 0 (Probability of impossible event is 0).
• For any event A, 0 ≤ P(A) ≤ 1.
• P(A') = 1 - P(A) (Probability of 'not A').
• P(A - B) = P(A ∩ B') = P(A) - P(A ∩ B).
• P(B - A) = P(B ∩ A') = P(B) - P(A ∩ B).
• Addition Theorem of Probability: For any two events A and B,
  P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
• If A and B are mutually exclusive (A ∩ B = ∅), then P(A ∩ B) = 0, and the addition theorem simplifies to:
  P(A ∪ B) = P(A) + P(B).
• For three events A, B, and C:
  P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C).
• If A₁, A₂, ..., A<0xE2><0x82><0x99> are mutually exclusive and exhaustive events, then P(A₁) + P(A₂) + ... + P(A<0xE2><0x82><0x99>) = P(A₁ ∪ A₂ ∪ ... ∪ A<0xE2><0x82><0x99>) = P(S) = 1.

6. Probability of Equally Likely Outcomes

If a sample space S consists of 'n' outcomes which are all equally likely, and an event E consists of 'm' of these outcomes (where m ≤ n), then the probability of event E is given by:

P(E) = (Number of outcomes favourable to E) / (Total number of possible outcomes) = m / n = n(E) / n(S)

• This is the classical definition of probability and is a direct consequence of the axiomatic approach when outcomes are equally likely.
• Key Skills: You often need to use Permutations and Combinations (Chapter 7) to calculate n(E) and n(S) in complex scenarios like drawing cards, selecting balls, arranging letters, etc.

Key Points for Competitive Exams:

• Understand the precise meaning of terms like 'mutually exclusive', 'exhaustive', 'at least one', 'exactly one', 'neither A nor B' (which is (A ∪ B)').
• Be comfortable applying the addition theorem: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
• Know how to calculate P(A') = 1 - P(A). This is often easier than calculating P(A) directly, especially for "at least one" type problems (calculate the probability of "none" and subtract from 1).
• Master counting techniques (Combinations and Permutations) to find n(E) and n(S) for problems involving selections or arrangements.
• Practice problems involving coins, dice, cards, balls in bags, etc., as they are standard templates.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts we've discussed. Try to solve them yourself first.

1. A single letter is selected at random from the word 'PROBABILITY'. What is the probability that it is a vowel?
   (A) 3/11
   (B) 4/11
   (C) 5/11
   (D) 2/11

2. Two dice are thrown simultaneously. What is the probability of getting a sum of 9?
   (A) 1/9
   (B) 1/6
   (C) 1/12
   (D) 5/36

3. If A and B are two events such that P(A) = 0.4, P(B) = 0.8 and P(B|A) = 0.6. Then P(A ∪ B) is equal to:
   (Note: P(B|A) = P(A ∩ B) / P(A))
   (A) 0.96
   (B) 0.24
   (C) 0.88
   (D) 0.12

4. Three coins are tossed. What is the probability of getting at least one head?
   (A) 1/8
   (B) 3/8
   (C) 5/8
   (D) 7/8

5. A bag contains 5 red and 3 blue balls. If 2 balls are drawn at random, what is the probability that both are red?
   (A) 5/14
   (B) 3/28
   (C) 10/28
   (D) 5/8

6. Let A and B be two mutually exclusive events of a random experiment such that P(not A) = 0.65 and P(A ∪ B) = 0.65. Find P(B).
   (A) 0.30
   (B) 0.35
   (C) 0.65
   (D) 0

7. If E is an event such that P(E) = 0.7, what is P(not E)?
   (A) 0.7
   (B) 0.3
   (C) 1
   (D) 0

8. Which of the following cannot be the probability of an event?
   (A) 2/3
   (B) -1.5
   (C) 15%
   (D) 0.7

9. From a standard deck of 52 cards, one card is drawn at random. What is the probability that it is either a king or a spade?
   (A) 17/52
   (B) 16/52
   (C) 4/52
   (D) 13/52

10. Two events A and B are said to be exhaustive if:
    (A) A ∩ B = ∅
    (B) A ∪ B = S
    (C) A ⊂ B
    (D) P(A) = P(B)

Answer Key:

1. (B) 4/11 (Vowels are O, A, I, I. Total letters = 11. Favourable = 4)
2. (A) 1/9 (Total outcomes = 36. Favourable outcomes for sum 9 are (3,6), (4,5), (5,4), (6,3) -> 4 outcomes. P = 4/36 = 1/9)
3. (C) 0.88 (P(A ∩ B) = P(B|A) * P(A) = 0.6 * 0.4 = 0.24. P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.8 - 0.24 = 1.2 - 0.24 = 0.96. Wait, rechecking calculation. P(A ∪ B) = 0.4 + 0.8 - 0.24 = 1.2 - 0.24 = 0.96. Let me re-evaluate the options or my calculation. Ah, I see the calculation error in the provided options/my initial check. Let's re-do: P(A ∩ B) = 0.24. P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.8 - 0.24 = 1.20 - 0.24 = 0.96. It seems option (A) should be correct. Let me double-check the question context - this uses conditional probability which is typically Class 12, but sometimes appears in competitive exams based on Class 11. Assuming the formula is given/known: P(A ∪ B) = 0.96. Let's assume there might be a typo in the question or options provided in the source I mentally referenced. If we assume P(A|B) = 0.6 instead, P(A ∩ B) = P(A|B)P(B) = 0.6 * 0.8 = 0.48. Then P(A ∪ B) = 0.4 + 0.8 - 0.48 = 1.2 - 0.48 = 0.72. This isn't an option either. Let's stick with the original interpretation P(B|A)=0.6 -> P(A ∩ B)=0.24 -> P(A ∪ B) = 0.96. Option (A) is 0.96. Correction: My initial check was correct, the answer should be 0.96, which is option A. Let me mark A. (A) 0.96
4. (D) 7/8 (Total outcomes = 2³ = 8 {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Only TTT has no heads. P(no heads) = 1/8. P(at least one head) = 1 - P(no heads) = 1 - 1/8 = 7/8)
5. (A) 5/14 (Total balls = 8. Ways to choose 2 balls = ⁸C₂ = (87)/(21) = 28. Ways to choose 2 red balls = ⁵C₂ = (54)/(21) = 10. P(both red) = 10/28 = 5/14)
6. (A) 0.30 (P(A) = 1 - P(not A) = 1 - 0.65 = 0.35. Since A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B). So, 0.65 = 0.35 + P(B). P(B) = 0.65 - 0.35 = 0.30)
7. (B) 0.3 (P(not E) = 1 - P(E) = 1 - 0.7 = 0.3)
8. (B) -1.5 (Probability must be between 0 and 1, inclusive. Negative probability is not possible).
9. (B) 16/52 (A = King, B = Spade. P(A) = 4/52. P(B) = 13/52. A ∩ B = King of Spades. P(A ∩ B) = 1/52. P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 4/52 + 13/52 - 1/52 = 16/52)
10. (B) A ∪ B = S (This is the definition of exhaustive events - their union covers the entire sample space).

Make sure you understand the reasoning behind each answer. Revise these concepts thoroughly. Good luck!