This MCQ module is based on: 14.1 Event
TOPIC 43 OF 45
14.1 Event
🎓 Class 11
Mathematics
CBSE
Theory
Ch 14 — Probability
⏱ ~15 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: 14.1 Event
Targeting Class 11 level in Probability, with Advanced difficulty.
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14.1 Event
In Class 9 you encountered probability through relative frequency: tossing a coin many times and noting that "heads" appears about half the time. We now formalise the language using sets — the foundation that lets probability scale from coin tosses to genome sequencing and weather forecasting.
A random experiment? is one whose outcome cannot be predicted with certainty even if the conditions are kept the same. Examples: tossing a coin, rolling a die, drawing a card from a deck, recording tomorrow's high temperature.
Sample space
The sample space \(S\) of a random experiment is the set of all possible outcomes.• Tossing one coin: \(S=\{H,T\}\).
• Tossing two coins: \(S=\{HH,HT,TH,TT\}\).
• Rolling a die: \(S=\{1,2,3,4,5,6\}\).
Each element of \(S\) is called a sample point.
Event
An event \(E\) is any subset of the sample space \(S\). The "occurrence" of \(E\) means the actual outcome lies in \(E\).
Example: Tossing two coins, \(S=\{HH,HT,TH,TT\}\). Some events:
| Description | Event \(\subseteq S\) |
|---|---|
| Number of tails is exactly 2 | \(A=\{TT\}\) |
| Number of tails is at least 1 | \(B=\{HT,TH,TT\}\) |
| Number of heads is at most 1 | \(C=\{HT,TH,TT\}\) |
| Second toss is not a head | \(D=\{HT,TT\}\) |
| Number of tails is more than two | \(\varnothing\) (impossible) |
14.1.1 Occurrence of an event
An event \(E\) is said to occur if the outcome \(\omega\) of the experiment is an element of \(E\). For instance, with \(E=\{2,4,6\}\) (even-die event), if you roll a 3 then \(E\) does not occur; if you roll a 4, \(E\) occurs.
14.1.2 Types of events
Six standard event types
- Impossible event: \(\varnothing\) (empty set). Never occurs.
- Sure / certain event: the whole sample space \(S\). Always occurs.
- Simple (elementary) event: a singleton \(\{ \omega\}\) — exactly one outcome.
- Compound event: an event containing more than one sample point.
- Complementary event: \(A'=S\setminus A\) (everything not in \(A\)).
- Mutually exclusive events: \(A\cap B=\varnothing\) — they cannot both occur.
14.1.3 Algebra of Events
Set-theory mapping
| In words | In set notation |
|---|---|
| Event "A or B" (at least one occurs) | \(A\cup B\) |
| Event "A and B" (both occur) | \(A\cap B\) |
| Event "A but not B" | \(A-B=A\cap B'\) |
| Event "neither A nor B" | \(A'\cap B'=(A\cup B)'\) |
| "\(A\) implies \(B\)" | \(A\subseteq B\) |
Venn diagram: \(A\cup B\) is the union of both shaded regions; \(A\cap B\) is the overlap.
Mutually exclusive and exhaustive events
Definitions
Events \(A_1, A_2, \ldots, A_n\) are
- mutually exclusive if \(A_i\cap A_j=\varnothing\) whenever \(i\ne j\) — no two can occur together;
- exhaustive if \(A_1\cup A_2\cup\cdots\cup A_n=S\) — at least one always occurs;
- mutually exclusive and exhaustive if both — they form a partition of the sample space.
Worked Examples
Example 1. Three coins are tossed. Describe the sample space.
\(S=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}\) — 8 equally likely outcomes.
Example 2. Two dice are rolled. (i) Describe the sample space. (ii) Write the event \(A\) "the sum of the two faces is 7".
(i) \(S=\{(i,j):1\le i,j\le 6\}\) has 36 outcomes.
(ii) \(A=\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}\) — 6 outcomes.
(ii) \(A=\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}\) — 6 outcomes.
Example 3. In rolling a single die, let \(A=\{1,3,5\}\) (odd) and \(B=\{2,4,6\}\) (even). Are A and B mutually exclusive? Exhaustive?
\(A\cap B=\varnothing\) — yes, mutually exclusive. \(A\cup B=\{1,2,3,4,5,6\}=S\) — yes, exhaustive. So \(A,B\) form a partition.
Example 4. A die is rolled. Let \(E=\{1,3,5\}, F=\{2,3\}, G=\{2,3,4,5\}\). Find \(E\cup F, E\cap F, F\cap G, F'\).
\(E\cup F=\{1,2,3,5\}\). \(E\cap F=\{3\}\). \(F\cap G=\{2,3\}\). \(F'=\{1,4,5,6\}\).
Example 5. Two dice. Let \(A=\) "sum is even", \(B=\) "sum is multiple of 3". Are they mutually exclusive? Are they exhaustive?
\(A\cap B\) = sum is even AND multiple of 3 = sum is 6 or 12 = \(\{(1,5),(2,4),(3,3),(4,2),(5,1),(6,6)\}\) — non-empty, so NOT mutually exclusive. \(A\cup B\) — sums of \(\{2,3,4,6,8,9,10,12\}\) — misses \(\{5,7,11\}\), so NOT exhaustive.
Example 6. A coin is tossed. If it shows tails, a die is thrown. Otherwise the coin is tossed again. Describe the sample space.
\(S=\{HH, HT, T1, T2, T3, T4, T5, T6\}\) — 8 outcomes (note: outcomes are NOT equally likely, but that's a probability concern, not a sample-space concern).
Activity: Build a Partition
L3 ApplyMaterials: Standard 52-card deck (or imagined), pen and paper.
Predict: Can you find 4 events whose union is the entire deck and which don't overlap pairwise?
- Sample space: 52-card deck. List 4 obvious partitions: by suit, by rank parity, by face/number, by color.
- Verify "by suit" (♠, ♥, ♦, ♣): 4 disjoint events of 13 cards each, total 52 = exhaustive.
- Try "by color" (red/black): only 2 events of 26 each — disjoint and exhaustive.
- Now check: is "diamond" mutually exclusive with "red"? Compute the intersection.
- Conclude: mutual exclusion and exhaustiveness are independent properties; check both.
"Diamond" ∩ "Red" = "diamond" (all diamonds are red), so they are NOT mutually exclusive. Diamonds ⊂ Reds. The partition by suit is finer than the partition by color: the suit-partition refines the color-partition. Recognising partitions is key to the addition law of probability that follows in Part 2.
Competency-Based Questions
Scenario: A QA engineer tests 3 components in sequence; each is "good" (G) or "defective" (D).
Q1. Number of outcomes in the sample space:
L1 RememberAnswer: (c) 8. \(2^3=8\) outcomes (each of 3 trials has 2 results).
Q2. Event A = "exactly one defective". Write A as a subset of S.
L3 ApplyAnswer: A = {DGG, GDG, GGD} — three outcomes (the defective can be in position 1, 2, or 3).
Q3. (T/F) "Events 'no defectives' and 'all defectives' are mutually exclusive AND exhaustive." Justify.
L5 EvaluateFalse. They are mutually exclusive (\(\{GGG\}\cap\{DDD\}=\varnothing\)) but NOT exhaustive (their union misses outcomes like DGG, GDD, etc.). Mutual exclusion does not imply exhaustiveness.
Q4. Apply: write the event "at least 2 defective" using union of simpler events. List its outcomes.
L3 ApplyAnswer: "exactly 2 def." ∪ "exactly 3 def." = \(\{DDG,DGD,GDD\}\cup\{DDD\}=\{DDG,DGD,GDD,DDD\}\) — 4 outcomes.
Q5. Design: find a partition of S into 4 mutually exclusive and exhaustive events labelled by the number of defectives (0, 1, 2, 3). Verify all properties.
L6 CreateSolution: A₀={GGG}, A₁={DGG,GDG,GGD}, A₂={DDG,DGD,GDD}, A₃={DDD}. Sizes: 1+3+3+1=8 ✓. Pairwise disjoint (different defective counts) ✓. Union = S ✓. Note: 1, 3, 3, 1 is row 3 of Pascal's Triangle — a peek at the binomial probability distribution, coming in Class 12.
Assertion–Reason Questions
Assertion (A): The sample space of tossing 4 coins has 16 outcomes.
Reason (R): Each coin contributes 2 possibilities and they are independent, so \(|S|=2^4\).
Reason (R): Each coin contributes 2 possibilities and they are independent, so \(|S|=2^4\).
Answer: (a). R is the multiplication principle that yields A.
Assertion (A): If \(A\) and \(B\) are mutually exclusive, then \(A\cup B=S\).
Reason (R): Mutually exclusive means \(A\cap B=\varnothing\).
Reason (R): Mutually exclusive means \(A\cap B=\varnothing\).
Answer: (d). A is false: mutual exclusion does NOT imply exhaustiveness. R is true.
Assertion (A): An event and its complement are mutually exclusive and exhaustive.
Reason (R): \(A\cap A'=\varnothing\) and \(A\cup A'=S\).
Reason (R): \(A\cap A'=\varnothing\) and \(A\cup A'=S\).
Answer: (a). R is the literal definition that gives A.
Frequently Asked Questions
What is a sample space?
The sample space S of a random experiment is the set of all possible outcomes. Tossing a coin: S = {H, T}; tossing two coins: S = {HH, HT, TH, TT}.
What is an event?
An event is any subset of the sample space. In rolling a die, A = {2, 4, 6} is the event 'an even number appears'.
What is the difference between a simple and a compound event?
A simple event has exactly one outcome — like {3} when rolling a die. A compound event has more than one outcome — like {2, 4, 6}.
What does mutually exclusive mean?
Two events A and B are mutually exclusive if they cannot occur together: A ∩ B = ∅. Example: when rolling a die, 'getting an even number' and 'getting an odd number' are mutually exclusive.
What does exhaustive mean?
Events A₁, A₂, …, Aₙ are exhaustive if their union covers the entire sample space: A₁ ∪ A₂ ∪ … ∪ Aₙ = S. Together they describe all possibilities.
What is the complement of an event?
The complement A' of an event A is the set of all outcomes in S that are not in A. A and A' are mutually exclusive and exhaustive.
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