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Class 12 Statistics Notes · GSEB

Operations on Events

Probability — learn union, intersection, difference, mutually exclusive, and exhaustive events with interactive Venn diagrams. GSEB Class 12 Statistics notes.

Last updated: 22 Sep 2026

Notes

Operations on Events

Probability uses set operations to combine events. The key operations are union (∪), intersection (∩), difference (−), mutually exclusive, and exhaustive.

Union of Events (A ∪ B)

Union — A ∪ B
The event that A occurs, or B occurs, or both occur. Read as 'A or B'. A ∪ B contains all outcomes that belong to A, or to B, or to both.

General Addition Law

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

Samosa Stall Test

A student sells samosas at the college canteen. Event A = "sells 10+ samosas", Event B = "sells 5+ chai". Some customers buy both. When counting total customers served (A ∪ B), you must subtract those who bought both — otherwise you count them twice!
SAB

Intersection of Events (A ∩ B)

Intersection — A ∩ B
The event that BOTH A and B occur together. Read as 'A and B'. A ∩ B contains only the outcomes that are common to both A and B.

Intersection

P(A∩B)P(A \cap B)
Die roll: A = {1,2,3,4}, B = {3,4,5,6}

A ∩ B = {3,4} — outcomes common to both.

SAB

Difference of Events (A − B)

Difference — A − B
The event that A occurs but B does NOT occur. Also written as A \\ B. Contains outcomes in A that are NOT in B.

A − B

SAB

Formula 1

P(A−B)=P(A)−P(A∩B)P(A - B) = P(A) - P(A \cap B)

Formula 2

P(A−B)=P(A∪B)−P(B)P(A - B) = P(A \cup B) - P(B)

Outcomes in A only — the blue crescent.

B − A

SAB

Formula 1

P(B−A)=P(B)−P(A∩B)P(B - A) = P(B) - P(A \cap B)

Formula 2

P(B−A)=P(A∪B)−P(A)P(B - A) = P(A \cup B) - P(A)

Outcomes in B only — the amber crescent.

Die: A = {1,2,3,4}, B = {3,4,5,6}

A − B = {1,2} — in A but not in B.  |  B − A = {5,6} — in B but not in A.

Samosa Stall Test

Event A = "customer buys samosa", Event B = "customer buys chai". A − B = "bought samosa but NOT chai" (only samosa lovers). B − A = "bought chai but NOT samosa" (only chai drinkers). The intersection A ∩ B = "bought both" is excluded from both differences.

Mutually Exclusive Events

Mutually Exclusive
Two events A and B are mutually exclusive if they CANNOT happen together — A ∩ B = ∅. No outcome belongs to both events.

Condition

A∩B=∅A \cap B = \emptyset

Addition Law (simplified)

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

Mutually Exclusive

A coin toss: H and T cannot both appear. P(H ∪ T) = P(H) + P(T) = 1/2 + 1/2 = 1.

NOT Mutually Exclusive

Drawing a card: A = "heart", B = "king". The king of hearts is in both — A ∩ B ≠ ∅.
SAB

Exhaustive Events

Exhaustive Events
Events that together cover the entire sample space — at least one of them must occur. A₁ ∪ A₂ ∪ ... ∪ Aₖ = S.

Condition

A∪B=SA \cup B = S

Probability

P(A∪B)=1P(A \cup B) = 1

Die roll: A = {1,2,3}, B = {4,5,6}

A ∪ B = {1,2,3,4,5,6} = S — they are exhaustive.

SAB

Explore: Venn Diagram Operations

Toggle between union, intersection, A only, and B only. Try the "Mutually Exclusive" toggle to see the circles separate. Watch the formulas change with each mode.

Event Operations — Interactive Venn
SAB123456

A ∪ B

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

A = {1, 2, 3, 4}

B = {3, 4, 5, 6}

Active (A ∪ B) = {1, 2, 3, 4, 5, 6}

n = 6  |  P = 6/6 = 1.000

Key Takeaways

Key Takeaways

  • A ∪ B = outcomes in A or B (or both). P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
  • A ∩ B = outcomes common to both A and B.
  • A − B = outcomes in A but not in B. P(A−B) = P(A) − P(A∩B) = P(A∪B) − P(B).
  • Mutually exclusive: A ∩ B = ∅, so P(A ∪ B) = P(A) + P(B) — no overlap to subtract.
  • Exhaustive: A ∪ B = S, so P(A ∪ B) = 1 — together they cover everything.

Practice

  1. A = {1,3,5}, B = {2,4,6} on a die. Find A ∪ B, A ∩ B. Are they mutually exclusive? Exhaustive?
  2. In a class of 40 students, 25 play cricket and 15 play football. If 8 play both, how many play neither?
  3. Two dice are rolled. A = "sum is 7", B = "sum is 11". Are A and B mutually exclusive? Find P(A ∪ B).

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