Class 11 Mathematics Notes Chapter 16 (Probability) – Mathematics Book

Alright class, let's get straight into Chapter 16: Probability. This is a fundamental chapter, not just for your Class 11 exams, but it forms the bedrock for many quantitative aptitude sections in government exams. So, pay close attention.

We'll break down the concepts as presented in the NCERT textbook, focusing on clarity and application.

Chapter 16: Probability - Detailed Notes

1. Introduction
Probability is the measure of the likelihood or chance that an event will occur. It's quantified as a number between 0 and 1 (inclusive).

• 0 indicates impossibility.
• 1 indicates certainty.

2. Random Experiments
An experiment is called a random experiment if it satisfies two conditions:
* It has more than one possible outcome.
* It is not possible to predict the outcome in advance.

• Examples: Tossing a coin, rolling a die, drawing a card from a well-shuffled deck, selecting a ball from a bag.

3. Outcomes and Sample Space

• Outcome: A possible result of a random experiment.
  • Example: Getting 'Heads' when tossing a coin is an outcome. Getting '4' when rolling a die is an outcome.
• Sample Space (S): The set of all possible outcomes of a random experiment. It is denoted by 'S'.
  • Example 1 (Tossing a coin): S = {H, T} ; Number of outcomes, n(S) = 2
  • Example 2 (Rolling a die): S = {1, 2, 3, 4, 5, 6} ; n(S) = 6
  • Example 3 (Tossing two coins): S = {HH, HT, TH, TT} ; n(S) = 4
  • Example 4 (Rolling two dice): S = {(1,1), (1,2), ..., (1,6), (2,1), ..., (6,6)} ; n(S) = 36
  • Example 5 (Drawing a card from a deck of 52): n(S) = 52

4. Events

• Event (E): Any subset of a sample space S is called an event. It represents one or more outcomes of interest.
  • Example (Rolling a die):
    • Event A: Getting an even number. A = {2, 4, 6}. A is a subset of S = {1, 2, 3, 4, 5, 6}.
    • Event B: Getting a prime number. B = {2, 3, 5}. B is a subset of S.
• Types of Events:
  • Impossible Event: An event that cannot occur. It corresponds to the empty set (∅). P(∅) = 0.
    • Example: Getting a '7' when rolling a standard die.
  • Sure Event: An event that is certain to occur. It corresponds to the entire sample space (S). P(S) = 1.
    • Example: Getting a number less than 7 when rolling a standard die.
  • Simple (or Elementary) Event: An event consisting of only one outcome (a singleton subset of S).
    • Example: Getting 'Heads' {H} when tossing a coin. Getting { (1, 2) } when rolling two dice.
  • Compound Event: An event consisting of more than one outcome.
    • Example: Getting an even number {2, 4, 6} when rolling a die.

5. Algebra of Events
Since events are sets, we can perform set operations on them:

• Complementary Event ('not A'): Denoted by A' or Aᶜ. It represents the set of all outcomes in S that are not in A. A' = S - A.
  • Example (Rolling a die): If A = {2, 4, 6} (getting an even number), then A' = {1, 3, 5} (getting an odd number).
• Event 'A or B' (Union): Denoted by A ∪ B. It represents the set of all outcomes that are in A, or in B, or in both.
  • Example (Rolling a die): If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.
• Event 'A and B' (Intersection): Denoted by A ∩ B. It represents the set of all outcomes that are common to both A and B.
  • Example (Rolling a die): If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}.
• Event 'A but not B' (Difference): Denoted by A - B or A ∩ B'. It represents the set of all outcomes that are in A but not in B. A - B = A ∩ B'.
  • Example (Rolling a die): If A = {1, 2, 3} and B = {3, 4, 5}, then A - B = {1, 2}.

6. Mutually Exclusive Events
Two events A and B are said to be mutually exclusive if they cannot occur simultaneously. This means their intersection is the empty set.

• A ∩ B = ∅
• Example (Rolling a die): Event E = {1, 3, 5} (getting an odd number) and Event F = {2, 4, 6} (getting an even number). E ∩ F = ∅. So, E and F are mutually exclusive.
• Example (Drawing a card): Event A = Getting a King, Event B = Getting a Queen. A and B are mutually exclusive. Event C = Getting a Heart, Event D = Getting a King. C and D are not mutually exclusive because you can draw the King of Hearts (C ∩ D = {King of Hearts}).

7. Exhaustive Events
A set of events E₁, E₂, ..., Eₙ associated with a sample space S are said to be exhaustive if their union is the entire sample space S.

• E₁ ∪ E₂ ∪ ... ∪ Eₙ = S
• This means that at least one of these events must occur when the experiment is performed.
• Example (Rolling a die): Event E = {1, 3, 5} (odd) and Event F = {2, 4, 6} (even). E ∪ F = {1, 2, 3, 4, 5, 6} = S. So, E and F are exhaustive events.
• Note: Events can be mutually exclusive, exhaustive, both, or neither.

8. Axiomatic Approach to Probability
Let S be the sample space of a random experiment. The probability P is a real-valued function whose domain is the power set of S (the set of all subsets/events of S) and range is the interval [0, 1], satisfying 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 E and F are mutually exclusive events (E ∩ F = ∅), then P(E ∪ F) = P(E) + P(F).
* This extends to any finite or countably infinite number of pairwise mutually exclusive events.

9. Probability of an Event (Equally Likely Outcomes)
If a sample space S consists of 'n' equally likely outcomes, and an event E consists of 'm' of these outcomes (favourable outcomes), then the probability of event E is:

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

10. Important Probability Results

• P(∅) = 0 (Probability of impossible event is 0)
• 0 ≤ P(E) ≤ 1 (Probability lies between 0 and 1)
• Probability of Complement: P(A') = 1 - P(A) or P(not A) = 1 - P(A)
• Addition Rule for any two events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  • This is crucial. It accounts for the outcomes counted twice (in the intersection).
• Addition Rule for Mutually Exclusive Events: If A and B are mutually exclusive (A ∩ B = ∅), then P(A ∩ B) = 0, and the rule simplifies to P(A ∪ B) = P(A) + P(B).
• Probability of Difference: P(A - B) = P(A ∩ B') = P(A) - P(A ∩ B)
• De Morgan's Laws (in probability context):
  • P(A' ∪ B') = P((A ∩ B)') = 1 - P(A ∩ B)
  • P(A' ∩ B') = P((A ∪ B)') = 1 - P(A ∪ B)

Key Focus Areas for Government Exams:

• Calculating n(S) and n(E) accurately, especially for dice, coins, and cards problems. Sometimes basic permutation/combination is needed here.
• Understanding and applying the concepts of mutually exclusive and exhaustive events.
• Mastering the Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
• Using the Complement Rule: P(A') = 1 - P(A).
• Problems involving drawing balls from bags/urns.

Multiple Choice Questions (MCQs)

1. The probability of any event E lies in the range:
   a) 0 < P(E) < 1
   b) 0 ≤ P(E) ≤ 1
   c) -1 ≤ P(E) ≤ 1
   d) P(E) > 0

2. A coin is tossed three times. What is the total number of possible outcomes (size of the sample space)?
   a) 3
   b) 6
   c) 8
   d) 9

3. A single standard die is rolled. What is the probability of getting a number greater than 4?
   a) 1/6
   b) 1/3
   c) 1/2
   d) 2/3

4. If P(A) = 0.65, what is the probability of the event 'not A', i.e., P(A')?
   a) 0.35
   b) 0.45
   c) 0.65
   d) 1

5. Let A and B be two events such that P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2. What is P(A ∪ B)?
   a) 0.7
   b) 0.9
   c) 0.1
   d) 1.1

6. If A and B are mutually exclusive events, then:
   a) P(A ∪ B) = P(A) * P(B)
   b) P(A ∩ B) = 1
   c) P(A ∪ B) = P(A) + P(B)
   d) P(A) = P(B)

7. One card is drawn from a well-shuffled deck of 52 cards. What is the probability that the card drawn is a face card (King, Queen, Jack)?
   a) 1/13
   b) 4/13
   c) 3/13
   d) 1/4

8. 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) 4/9

9. Which of the following probabilities is not possible for an event?
   a) 2/3
   b) 0.01
   c) 1.05
   d) 0

10. If E is an event such that P(E) = 0, what type of event is E?
    a) Sure Event
    b) Simple Event
    c) Compound Event
    d) Impossible Event

Answer Key:

1. b) 0 ≤ P(E) ≤ 1
2. c) 8 (Outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. n(S) = 2³)
3. b) 1/3 (Favourable outcomes {5, 6}. n(E)=2. n(S)=6. P(E) = 2/6 = 1/3)
4. a) 0.35 (P(A') = 1 - P(A) = 1 - 0.65 = 0.35)
5. a) 0.7 (P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.5 - 0.2 = 0.7)
6. c) P(A ∪ B) = P(A) + P(B) (Because P(A ∩ B) = 0 for mutually exclusive events)
7. c) 3/13 (There are 3 face cards per suit x 4 suits = 12 face cards. P(Face Card) = 12/52 = 3/13)
8. a) 1/9 (Favourable outcomes for sum 9 are (3,6), (4,5), (5,4), (6,3). n(E)=4. n(S)=36. P(E) = 4/36 = 1/9)
9. c) 1.05 (Probability cannot be greater than 1)
10. d) Impossible Event (An event with probability 0 cannot occur)

Make sure you understand the reasoning behind each answer. Practice more problems, especially involving combinations of these rules. Good luck!