Event and Its Types – Class 11 Applied Mathematics

Basic Definitions and Sample Space

An experiment is a process that results in a well-defined outcome. The set of all possible outcomes of a random experiment is called the sample space, denoted by S. An event E is a subset of the sample space S, representing a specific outcome or a collection of outcomes.

n(E) ≤ n(S)

Example: Solved Example: Rolling a die

Show Step-by-Step Solution

Step 1: Identify sample space S = {1, 2, 3, 4, 5, 6}.
Step 2: Define event E as getting an even number, E = {2, 4, 6}.
Step 3: Calculate n(E) = 3.
Answer: The event E has 3 outcomes.

Types of Events: Simple and Compound

A simple event (or elementary event) contains exactly one sample point from the sample space. A compound event contains more than one sample point. For example, in tossing two coins, {HH} is a simple event, while {at least one head} is a compound event.

n(E) = 1 (Simple Event)

Example: Solved Example: Tossing two coins

Show Step-by-Step Solution

Step 1: S = {HH, HT, TH, TT}.
Step 2: Simple event E₁ = {HH}.
Step 3: Compound event E₂ = {HH, HT, TH}.
Answer: E₁ is simple, E₂ is compound.

Mutually Exclusive and Exhaustive Events

Two events A and B are mutually exclusive if A ∩ B = ∅, meaning they cannot occur simultaneously. Events are collectively exhaustive if their union equals the sample space S, meaning at least one of them must occur.

A ∩ B = ∅ (Mutually Exclusive); A ∪ B = S (Exhaustive)

Example: Solved Example: Verify mutual exclusivity

Show Step-by-Step Solution

Step 1: Let S = {1, 2, 3, 4, 5, 6}.
Step 2: Let A = {1, 2} and B = {4, 5}.
Step 3: A ∩ B = ∅.
Answer: A and B are mutually exclusive.

Section A: Multiple Choice Questions (MCQs)

easyIf S = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}, find n(A).

A. 2 B. 3 C. 4 D. 6

Show Answer

✔ Correct: B

The set A contains elements {2, 4, 6}, so the count n(A) = 3.

easyIn a single toss of a coin, S = {H, T}. What is the event of getting a Head?

A. {H} B. {T} C. {H, T} D. ∅

Show Answer

✔ Correct: A

The event of getting a head is the subset containing only the outcome H.

mediumIf A = {1, 3, 5} and B = {2, 4, 6} are subsets of S = {1, 2, 3, 4, 5, 6}, what is A ∪ B?

A. {1, 2, 3} B. {4, 5, 6} C. {1, 2, 3, 4, 5, 6} D. ∅

Show Answer

✔ Correct: C

The union of A and B combines all unique elements from both sets, resulting in {1, 2, 3, 4, 5, 6}.

mediumIf A = {1, 2} and B = {3, 4} in S = {1, 2, 3, 4}, are they mutually exclusive?

A. Yes, because A ∩ B = ∅ B. No, because A ∩ B = {1, 2} C. Yes, because A ∪ B = S D. No, because A ∪ B = ∅

Show Answer

✔ Correct: A

Since A and B share no common elements, their intersection is the empty set, making them mutually exclusive.

hardA die is rolled. Event A = {x : x is prime}, Event B = {x : x is even}. Find A ∩ B.

A. {2} B. {3} C. {2, 3} D. {2, 4, 6}

Show Answer

✔ Correct: A

A = {2, 3, 5}, B = {2, 4, 6}. The intersection A ∩ B = {2}.

hardIf P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2, find P(A ∪ B).

A. 0.7 B. 0.9 C. 0.1 D. 1.1

Show Answer

✔ Correct: A

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.5 - 0.2 = 0.7.

very hardTwo dice are rolled. Let E be the event that the sum is 10. Find n(E).

A. 2 B. 3 C. 4 D. 5

Show Answer

✔ Correct: B

Possible pairs for sum 10 are (4, 6), (5, 5), (6, 4). Total count is 3.

Section B: Assertion–Reason Questions

Assertion (A): If A = {1, 2} and B = {2, 3}, then A and B are mutually exclusive.
Reason (R): Two events are mutually exclusive if their intersection is the empty set.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: D

The assertion is false because A ∩ B = {2} ≠ ∅. The reason is a correct definition.

Assertion (A): For a fair die, the event E = {1, 3, 5} and F = {2, 4, 6} are exhaustive.
Reason (R): Events are exhaustive if their union is the sample space S.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

E ∪ F = {1, 2, 3, 4, 5, 6} = S. Both assertion and reason are true and correctly linked.

Section C: Short Answer Questions

mediumGiven S = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3}, B = {3, 4, 5, 6}. Find: (i) A ∩ B (ii) A ∪ B (iii) A' (complement of A).

Show Answer & Solution

(i) A ∩ B = {3}
(ii) A ∪ B = {1, 2, 3, 4, 5, 6}
(iii) A' = {4, 5, 6, 7, 8}

mediumA coin is tossed three times. S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Find: (i) Event A: exactly two heads (ii) Event B: no heads (iii) Event C: at least one tail.

Show Answer & Solution

(i) A = {HHT, HTH, THH}
(ii) B = {TTT}
(iii) C = {HHT, HTH, THH, HTT, THT, TTH, TTT}

hardIf P(A) = 0.3, P(B) = 0.4, P(A ∩ B) = 0.1. Find: (i) P(A ∪ B) (ii) P(A') (iii) P(B').

Show Answer & Solution

(i) P(A ∪ B) = 0.3 + 0.4 - 0.1 = 0.6
(ii) P(A') = 1 - 0.3 = 0.7
(iii) P(B') = 1 - 0.4 = 0.6

Section D: Long Answer Questions

hardIn a survey of 20 students, 12 like Mathematics, 10 like Science, and 5 like both. Let M be the event a student likes Math and S be the event a student likes Science. (i) Find n(M ∪ S) (ii) Are M and S mutually exclusive? (iii) Find the number of students who like only Math.

Show Answer & Solution

Step 1: Given n(M) = 12, n(S) = 10, n(M ∩ S) = 5.
Step 2: Use formula n(M ∪ S) = n(M) + n(S) - n(M ∩ S).
Step 3: n(M ∪ S) = 12 + 10 - 5 = 17.
Step 4: Since n(M ∩ S) = 5 ≠ 0, they are not mutually exclusive.
Step 5: Only Math = n(M) - n(M ∩ S) = 12 - 5 = 7.

very hardA bag contains 5 red, 3 blue, and 2 green balls. Two balls are drawn. (i) Define the sample space size (ii) Find event E₁: both are red (iii) Find event E₂: one red and one blue.

Show Answer & Solution

Step 1: Total balls = 5 + 3 + 2 = 10. Total ways to draw 2 = ¹⁰C₂ = 45.
Step 2: Ways to draw 2 red = ⁵C₂ = 10.
Step 3: Ways to draw 1 red and 1 blue = ⁵C₁ × ³C₁ = 5 × 3 = 15.
Step 4: E₁ = 10/45 = 2/9.
Step 5: E₂ = 15/45 = 1/3.

Section E: Case-Based Question

★

Case Study: A school club has 50 members. 30 members participate in Debate, 20 participate in Quiz, and 10 participate in both. Let D be the event of choosing a Debate participant and Q be the event of choosing a Quiz participant.

Sub-question 1: (i) How many members participate in either Debate or Quiz? (ii) How many participate in neither?

Show Solution

(i) n(D ∪ Q) = n(D) + n(Q) - n(D ∩ Q) = 30 + 20 - 10 = 40.
(ii) Neither = Total - n(D ∪ Q) = 50 - 40 = 10.

Sub-question 2: If a member is chosen at random, what is the probability they participate in only Debate?

Show Solution

Only Debate = n(D) - n(D ∩ Q) = 30 - 10 = 20. Probability = 20/50 = 0.4.

CBSE Board & Sample Paper Past Year Questions

📋 CBSE 2023 📝 1 Mark

If P(A) = 0.5, P(B) = 0.3 and A, B are mutually exclusive, find P(A ∪ B).

Show Answer & Solution

Since they are mutually exclusive, P(A ∪ B) = P(A) + P(B) = 0.5 + 0.3 = 0.8.

📋 CBSE 2022 📝 1 Mark

A die is thrown. Find the probability of getting a number greater than 4.

Show Answer & Solution

S = {1, 2, 3, 4, 5, 6}. E = {5, 6}. P(E) = 2/6 = 1/3.

📋 CBSE Sample Paper 2024 📝 1 Mark

If A = {1, 2, 3}, B = {3, 4, 5}, find A ∩ B.

Show Answer & Solution

The common element is 3. So, A ∩ B = {3}.

📋 CBSE 2020 📝 1 Mark

If P(A) = 0.6, find P(A').

Show Answer & Solution

P(A') = 1 - P(A) = 1 - 0.6 = 0.4.

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