Topics
Number Systems
Real Numbers
Algebra
Polynomials
Coordinate Geometry
Pair of Linear Equations in Two Variables
- Pair of Linear Equations in Two Variables
- Graphical Method with Different Cases of Solution
- Algebraic Methods of Solving a Pair of Linear Equations
- Substitution Method
- Elimination Method
Geometry
Quadratic Equations
Trigonometry
Arithmetic Progressions
Coordinate Geometry
Mensuration
Statistics and Probability
Triangles
Circles
Introduction to Trigonometry
Heights and Distances
- Angles of Elevation and Depression
- Problems based on Elevation and Depression
Areas Related to Circles
Surface Areas and Volumes
Statistics
Probability
Estimated time: 3 minutes
Maharashtra State Board: Class 10, 12
CISCE: Class 10
CISCE: Class 10
Key Points: Types of Events in Probability
| Type of Event | Meaning | Probability |
|---|---|---|
| Sure (Certain) Event | An event that is certain to occur | P(E) = 1 |
| Impossible Event | An event that cannot occur | P(E) = 0 |
| Simple (Elementary) Event | An event having only one outcome | P(E) = 1 / n(S) |
| Complementary Event (E̅) | An event that occurs when E does not occur | P(not E) = 1 − P(E) |
| Mutually Exclusive Events | Two events that cannot occur together | P(A ∩ B) = 0 |
| Exhaustive Events | Events which together cover all outcomes of S | P(A₁) + P(A₂) + … = 1 |
| Equally Likely Events | All outcomes have the same chance of occurring | P(E) = n(E) / n(S) |
| General Rule | Probability of any event | 0 ≤ P(E) ≤ 1 |
Properties:
- Complement Rule
P(A′) = 1 − P(A)
⇒ P(A) + P(A′) = 1 - Range of Probability
0 ≤ P(A) ≤ 1 - Impossible Event
P(ϕ) = 0 - Certain Event
P(S) = 1 - Subset Rule
If A ⊆ B, then P(A) ≤ P(B) - Difference of Events
P(A ∩ B′) = P(A) − P(A ∩ B)
P(A′ ∩ B) = P(B) − P(A ∩ B) - Union of Two Events
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) - Union of Three Events
P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
− P(A ∩ B) − P(B ∩ C) − P(C ∩ A) + P(A ∩ B ∩ C) - Mutually Exclusive Events (2 events)
If A ∩ B = 0, then
P(A ∪ B) = P(A) + P(B) - Mutually Exclusive Events (multiple)
P(A₁ ∪ A₂ ∪ ... ∪ Aₙ) = P(A₁) + P(A₂) + ... + P(Aₙ) - Upper Bound of Union
P(A ∪ B) ≤ P(A) + P(B)
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