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Class 11 Maths Notes

Probability Class 11 Notes

Complete, exam-ready notes on probability: random experiments and outcomes, sample spaces and events, the axiomatic approach to probability, complementary events, the addition theorem, mutually exclusive events, and odds in favour and against — written for CBSE boards and JEE/NEET revision.

Class11SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is probability in one line?

Probability measures the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain), calculated as the ratio of favourable outcomes to total outcomes in the sample space.

Random Experiments and Outcomes

Random experiment

A process whose result is not determined in advance and can be one of several possible outcomes. Rolling a die, tossing a coin, or drawing a card from a deck are all random experiments.

  • An outcome is a single result of a random experiment.
  • A trial is one performance of the experiment.
  • Events are collections of one or more outcomes.

Sample Space and Events

Sample space (S)

The set of all possible outcomes of a random experiment. For a coin toss, S = {H, T}; for a die, S = {1, 2, 3, 4, 5, 6}. A sample space is exhaustive (covers all outcomes) and mutually exclusive (no two outcomes occur simultaneously).

  • An event is a subset of the sample space.
  • A sure event is the entire sample space S; probability = 1.
  • An impossible event is the empty set ∅; probability = 0.

Types of Events

  • Simple (elementary) event: contains exactly one outcome.
  • Compound event: contains two or more outcomes.
  • Complementary event A': the event that A does not occur.
  • Mutually exclusive events: cannot occur together;
  • A∩B=∅A \cap B = \emptyset
  • .
  • Exhaustive events: their union covers the entire sample space.

Quick check

If events are both mutually exclusive and exhaustive, their probabilities sum to 1. This is the basis for the axiomatic definition of probability.

Axioms of Probability

  • Axiom 1:
  • 0≤P(A)≤10 \leq P(A) \leq 1
  • for every event A.
  • Axiom 2:
  • P(S)=1P(S) = 1
  • (probability of the sure event is 1).
  • Axiom 3: For mutually exclusive events,
  • P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
  • .
  • Classical definition: if all n outcomes are equally likely and m are favourable,
  • P(A)=mnP(A) = \frac{m}{n}
  • .
P(A)=n(A)n(S)P(A) = \frac{n(A)}{n(S)}
Classical probability

Complementary Events and the Addition Theorem

Addition theorem

For any two events A and B, the probability that at least one occurs is: P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B). The intersection term is subtracted because outcomes in both A and B are counted twice.

  • Complementary event:
  • P(A′)=1−P(A)P(A') = 1 - P(A)
  • .
  • If A and B are mutually exclusive,
  • P(A∩B)=0P(A \cap B) = 0
  • and the formula simplifies to
  • P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
  • .
  • For three events:
  • P(A∪B∪C)=P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)P(A \cup B \cup C) = P(A)+P(B)+P(C)-P(A \cap B)-P(A \cap C)-P(B \cap C)+P(A \cap B \cap C)
  • .

Mutually Exclusive Events

Mutually exclusive (disjoint) events

Two events are mutually exclusive if they cannot happen at the same time: A∩B=∅A \cap B = \emptyset, so P(A∩B)=0P(A \cap B) = 0. Rolling a 3 and rolling a 5 on a single die are mutually exclusive.

  • Mutually exclusive events are NOT the same as independent events.
  • For mutually exclusive events,
  • P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
  • — no intersection to subtract.
  • If events are both mutually exclusive and exhaustive, their probabilities sum to 1.

Odds in Favour and Against

Odds

Odds in favour of event A are the ratio P(A):P(A′)P(A) : P(A'). If P(A) = 3/5, odds in favour are 3:2. Odds against A are P(A′):P(A)P(A') : P(A) — the reverse ratio.

  • Odds in favour = m:(n − m) if P(A) = m/n.
  • Odds against = (n − m):m.
  • If odds in favour are a:b, then
  • P(A)=aa+bP(A) = \frac{a}{a+b}
  • .
  • Converting between probability and odds is a common JEE short-cut.

Solved Examples

Example: Two dice are thrown. Find the probability that the sum is 7.

Solution: Total outcomes = 6 × 6 = 36. Favourable pairs summing to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 outcomes. Probability = 6/36 = 1/6.

Example: From a pack of 52 cards, one card is drawn. Find the probability that it is either a king or a heart.

Solution: P(king) = 4/52, P(heart) = 13/52, P(king of hearts) = 1/52. By the addition theorem, P(king or heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Classical probability

P(A)=n(A)n(S)P(A) = \frac{n(A)}{n(S)}

Probability bounds

0≤P(A)≤10 \leq P(A) \leq 1

Complementary event

P(A′)=1−P(A)P(A') = 1 - P(A)

Addition theorem

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Mutually exclusive intersection

P(A∩B)=0P(A \cap B) = 0

Mutually exclusive addition

P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)

Odds in favour

Odds in favour=P(A):P(A′)\text{Odds in favour} = P(A) : P(A')

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • Classical probability assumes equally likely outcomes — always verify this first.
  • Mutually exclusive means P(A ∩ B) = 0; it does NOT mean independent.
  • The addition theorem subtracts the intersection to avoid double-counting.
  • Complementary events: P(A') = 1 − P(A) — useful when counting favourable outcomes is harder.
  • Odds in favour a:b means P(A) = a/(a+b) — a common conversion in exams.
  • For three events, include the triple intersection term in the addition formula.
  • The sample space must be exhaustive and mutually exclusive.
  • Dice problems: total outcomes = 36; card problems: total = 52.

FAQ

Common questions

What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot occur together (P(A ∩ B) = 0); independent events do not affect each other's probability (P(A ∩ B) = P(A)·P(B)). They are distinct concepts.

How do you convert probability to odds?

If P(A) = p, odds in favour = p:(1−p). If odds are a:b, then P(A) = a/(a+b).

When do I use the addition theorem?

Use it when you need the probability of A or B (at least one occurring). Subtract P(A ∩ B) to avoid counting shared outcomes twice.

What is a sure event?

A sure event is the entire sample space S — it includes every possible outcome. Its probability is always 1.

Continue learning

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