Where Mindful Learning Begins
Where Mindful Learning Begins
Every question from Exercise 14.1, Exercise 14.2 and the Miscellaneous Exercise, solved step by step — describing events with set notation, and assigning and computing probabilities the axiomatic way.
Probability builds directly on the Sets and the Permutations & Combinations chapters. Every event is treated as a subset of a sample space, so "A or B", "A and B" and "not A" translate exactly into union, intersection and complement. The chapter then classifies events as mutually exclusive, exhaustive, or both, before laying down the axiomatic definition of probability — a small set of rules any valid probability assignment must satisfy — and building the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) on top of it.
This Class 11 Maths NCERT Solutions Chapter 14 hub covers Probability — the chapter that formalises "chance" using the language of sets. Every event associated with a random experiment is defined as a subset of the sample space S. Simple events have exactly one outcome, compound events have more than one, the empty set φ is the impossible event, and S itself is the sure event.
Two or more events combine using the same set operations from the Sets chapter: 'A or B' is A∪B, 'A and B' is A∩B, 'A but not B' is A−B, and 'not A' is the complement A′. Two events are mutually exclusive if they cannot both occur (A∩B=φ), and a collection of events is exhaustive if their union covers the whole sample space — together, mutually exclusive and exhaustive events partition S completely.
The axiomatic approach then defines probability as any function P satisfying three rules — P(E) ≥ 0, P(S) = 1, and P(E∪F) = P(E)+P(F) for mutually exclusive E and F — which is more general than assuming every outcome is equally likely. Exercise 14.1 (7 questions) is entirely about describing and classifying events. Exercise 14.2 (21 questions) covers valid probability assignments, equally likely outcomes, and the addition and complement rules. The Miscellaneous Exercise (10 questions) applies all of this to card, marble, lottery and seating-arrangement problems that also draw on counting techniques from Permutations and Combinations.
An event is any subset of the sample space S. Simple, compound, impossible (φ) and sure (S) events. Exercise 14.1.
'A or B' = A∪B, 'A and B' = A∩B, 'A but not B' = A−B, 'not A' = A′. Exercise 14.1.
A and B are mutually exclusive if A∩B=φ; events are exhaustive if their union equals S. Exercise 14.1.
A valid probability assignment needs P(E)≥0 for every event, P(S)=1, and additivity over mutually exclusive events. Exercise 14.2.
P(A∪B)=P(A)+P(B)−P(A∩B) and P(not A)=1−P(A), applied to cards, dice, marbles and more. Exercise 14.2 and the Miscellaneous Exercise.
Everything you need before you start solving. This is a summary for quick recall — the Formula Cards above have the full printable version.
The event that A does not occur.
Occurs whenever A occurs, B occurs, or both do.
Both translate directly from the corresponding set operations.
A and B cannot occur simultaneously — the corresponding sets are disjoint.
At least one of the events is guaranteed to occur on every performance of the experiment.
Any assignment of numbers to outcomes satisfying all three is a valid probability — outcomes need not be equally likely.
Number of outcomes favourable to A, divided by the total number of outcomes — valid only when every outcome is equally likely.
Corrects for outcomes in A∩B being counted twice.
Valid only when A∩B=φ, so the correction term vanishes.
Since A and A′ are mutually exclusive and exhaustive.
| If you see... | Use this approach |
|---|---|
| Asked to describe "A or B", "A and B", "A but not B", or "not A" as a set | Translate directly: A∪B, A∩B, A−B (= A∩B′), and A′ = S−A. |
| Asked whether two events are mutually exclusive | List both events as sets and check whether A∩B=φ (no outcome in common). |
| Asked whether a collection of events is exhaustive | Check whether the union of all the events equals the full sample space S. |
| Given a table of proposed probabilities for each outcome | Check every value is ≥ 0 and that all the values sum to exactly 1 — both conditions must hold. |
| Outcomes are stated (or clearly implied) to be equally likely | Use P(A) = n(A)/n(S) — count favourable outcomes over total outcomes. |
| Asked for P(A or B) and A, B might overlap | Use the addition rule P(A∪B) = P(A) + P(B) − P(A∩B); only drop the last term if A∩B=φ. |
| Asked for the probability that an event does NOT happen | Use P(not A) = 1 − P(A) rather than computing the complement's outcomes directly. |
Drawn from where students actually lose marks across all three exercises.
Events — describing events as sets, the algebra of events, and identifying mutually exclusive and exhaustive events · 7 questions
Solve Exercise 14.1 →Axiomatic Approach to Probability — valid probability assignments, equally likely outcomes, and the addition and complement rules · 21 questions
Solve Exercise 14.2 →Mixed practice — cards, marbles, lotteries, seating and letter-envelope problems combining probability with counting techniques · 10 questions
Solve Miscellaneous →
Expert CBSE Coaching · Class 9–12