Schooling · Class 11 · Mathematics · Probability · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Probability: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- random experiment
- sample space S
- event
- mutually exclusive
- complement
- Axiomatic definition
- P(A) ≥ 0
- Formulas and facts
- 0 ≤ P(A) ≤ 1
- P(S) = 1
- P(∅) = 0
- P(A′) = 1 − P(A)
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Formula / Fact
- Worked examples
- Listing sample space and finding a…
- Using the addition rule
- Using the complement
- Watch out for
- Forgetting to subtract the…
- Treating events that can happen…
- Writing probability greater than 1…
- Confusing the complement of an event…
- Mixing up ordered and unordered…
What this chapter is about
Probability is the branch of mathematics that measures how likely an event is to occur. In Class 11, you study the axiomatic approach to probability, which builds the subject on a small set of clearly stated rules (axioms) rather than on intuitive ideas alone. This framework lets you handle experiments whose outcomes are equally likely, as well as those that are not.
Key ideas
- A random experiment is a process whose outcome cannot be predicted with certainty but whose set of all possible outcomes is known. Examples: rolling a die, drawing a card from a shuffled pack.
- The sample space S is the set of all possible outcomes of a random experiment. Each element of S is called a sample point.
- An event is any subset of the sample space. The event occurs if the actual outcome belongs to that subset.
- Two events are mutually exclusive (disjoint) if they cannot occur together; their intersection is empty.
- The complement of an event A, written A′ (or Aᶜ), contains every outcome in S that is not in A.
- Axiomatic definition: For every event A in sample space S, the probability P(A) satisfies three axioms:
- P(A) ≥ 0.
- P(S) = 1.
- If A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B).
- Equally likely outcomes: When every sample point has the same chance, P(A) = (number of outcomes favourable to A) / (total number of outcomes in S).
- The addition rule for any two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Formulas and facts to remember
- Formula / Fact: 0 ≤ P(A) ≤ 1 · Meaning: A probability is always between 0 and 1 inclusive.
- Formula / Fact: P(S) = 1 · Meaning: The certain event has probability 1.
- Formula / Fact: P(∅) = 0 · Meaning: The impossible event has probability 0.
- Formula / Fact: P(A′) = 1 − P(A) · Meaning: Probability of complement equals 1 minus probability of the event.
- Formula / Fact: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) · Meaning: General addition rule for two events.
- Formula / Fact: If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B) · Meaning: Special case; the intersection is empty.
- Formula / Fact: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(B ∩ C) − P(A ∩ C) + P(A ∩ B ∩ C) · Meaning: Inclusion-exclusion for three events.
Common mistakes
- Forgetting to subtract the intersection when using the addition rule → Always check whether A ∩ B is empty; if not, subtract P(A ∩ B).
- Treating events that can happen together as mutually exclusive → Events are mutually exclusive only when they share no common outcome.
- Writing probability greater than 1 or less than 0 → Recheck your counts; a valid probability lies between 0 and 1.
- Confusing the complement of an event with just one alternative outcome → The complement includes every outcome in S that is not in the event.
- Mixing up ordered and unordered counting → Use permutations when order matters, combinations when it does not.
Quick revision
- Sample space = set of all possible outcomes; event = any subset of the sample space.
- P(A) always lies between 0 and 1; P(S) = 1 and P(∅) = 0.
- P(A′) = 1 − P(A) — often the quickest route to "at least one" problems.
- Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Mutually exclusive events have no common outcome, so P(A ∩ B) = 0.
- For equally likely outcomes, P(A) = favourable outcomes / total outcomes.