Probability

Basic Terminology

Experiment

• An operation which results in some well-defined outcomes is called an experiment.

Random Experiment

• An experiment whose outcome cannot be predicted with certainty.
• Examples: Tossing a coin, throwing a die.

Sample Space

• The set of all possible outcomes of an experiment.
• Denoted by S.
• Example: Throwing a die → S = {1,2,3,4,5,6}

Event

• Any subset of the sample space.
• Denoted by E.

Types of Events

• Impossible (Null) Event: No outcome (ϕ)
• Sure Event: Whole sample space S
• Simple (Elementary) Event: Only one outcome
• Compound Event: More than one outcome
• Mutually Exclusive Events: E₁ ∩ E₂ = ϕ
• Mutually Exclusive & Exhaustive Events: Union of all events = S

Complement of an Event

• Complement of event E is denoted by E′ or E^c.
• E′ = S − E
• E ∪ E′ = S
• E ∩ E′ = ϕ

Probability of an Event

• P(E) = n(E) / n(S)
• 0 ≤ P(E) ≤ 1
• P(S) = 1
• P(ϕ) = 0
• If A ⊆ B ⇒ P(A) ≤ P(B)
• P(E) + P(E′) = 1

Odds

Odds in Favour of an Event

• Odds in favour of E = P(E) / P(E′)
• If odds in favour are a : b, then P(E) = a / (a + b)

Odds Against an Event

• Odds against E = P(E′) / P(E)
• If odds against are c : d, then P(E) = d / (c + d)

Standard Card Results (Pack of 52 Cards)

• Total cards = 52
• Red cards = 26 (Hearts 13, Diamonds 13)
• Black cards = 26 (Spades 13, Clubs 13)
• Aces = 4, Kings = 4, Queens = 4, Jacks = 4
• Face cards = 12 (Kings, Queens, Jacks)
• Honour cards = 16 (A, K, Q, J)

Other Important Types of Events

• Independent Events: Occurrence of one does not affect the other
• Dependent Events: Occurrence of one affects the other
• Equally Likely Events: All outcomes have equal probability

Addition Theorem of Probability

• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• If A and B are mutually exclusive:
• P(A ∪ B) = P(A) + P(B)
• For three events A, B, C:
• P(A ∪ B ∪ C) = P(A)+P(B)+P(C) − P(A∩B) − P(B∩C) − P(A∩C) + P(A∩B∩C)

Useful Probability Results

• P(A − B) = P(A) − P(A ∩ B)
• P(B − A) = P(B) − P(A ∩ B)
• P(A ∪ B) ≤ P(A) + P(B)
• P((A ∪ B)′) = 1 − P(A ∪ B)
• P((A ∩ B)′) = 1 − P(A ∩ B)

Conditional Probability

• P(A | B) = P(A ∩ B) / P(B), where P(B) ≠ 0
• Represents probability of A when B has already occurred

Multiplication Theorem

• P(A ∩ B) = P(A) · P(B | A)
• P(A ∩ B) = P(B) · P(A | B)
• If A and B are independent:
• P(A ∩ B) = P(A) · P(B)

Bayes’ Theorem

• If E₁, E₂, …, Eₙ are mutually exclusive and exhaustive events:
• P(Eᵢ | A) = [P(Eᵢ) · P(A | Eᵢ)] / [Σ P(Eⱼ) · P(A | Eⱼ)]

Binomial Probability Distribution

• n = number of trials
• p = probability of success
• q = probability of failure, p + q = 1
• P(X = r) = ⁿCᵣ · pʳ · qⁿ⁻ʳ

Mean & Variance of Binomial Distribution

• Mean = np
• Variance = npq
• Standard Deviation = √(npq)

JEE Main Focus Tips

• Card and coin problems are very common
• Always check whether events are independent or dependent
• Use complements to simplify calculations
• Bayes’ theorem questions are direct and scoring
• Binomial distribution is frequently tested