Probability

Event

An **event** is any subset E of the sample space S of a random experiment.

**Key Understanding:**

Every subset of S, including the empty set φ and S itself, is an event

Events correspond to outcomes and groups of outcomes we're interested in

Example: In tossing a coin twice, S = {HH, HT, TH, TT}. The event "exactly one head" is E = {HT, TH}

**Occurrence of an Event:**

An event E is said to have **occurred** if the outcome ω of the experiment satisfies ω ∈ E. If ω ∉ E, the event has not occurred.

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Types of Events

Impossible and Sure Events

**Impossible Event:** The empty set φ (no outcome satisfies the event condition)

Example: Rolling a die and getting a multiple of 7 → E = φ

**Sure Event:** The entire sample space S (every outcome ensures the event)

Example: Rolling a die and getting either odd or even → E = S

Simple (Elementary) Event

**Definition:** An event containing exactly one sample point.

**Properties:**

A sample space with n distinct elements has exactly n simple events

Example: Tossing two coins → E₁ = {HH}, E₂ = {HT}, E₃ = {TH}, E₄ = {TT} are four simple events

Compound Event

**Definition:** An event with more than one sample point.

**Examples:**

Tossing a coin thrice: "Exactly one head" = {HTT, THT, TTH}

Tossing a coin thrice: "At least one head" = {HTT, THT, TTH, HHT, HTH, THH, HHH}

Each contains multiple outcomes, hence compound.

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Algebra of Events

Events can be combined using set operations similar to combining sets.

Complementary Event (Not A)

**Definition:** A′ represents the event "not A"

**Formula:**

A′ = {ω : ω ∈ S and ω ∉ A} = S – A

**Property:** If A occurs, A′ doesn't occur, and vice versa

**Example:** Tossing three coins, S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

If A = "only one tail" = {HHH, HHT, HTH}, then A′ = {HTT, THT, TTH, TTT}

Event 'A or B' (Union)

**Definition:** A ∪ B represents "A or B or both"

**Formula:**

A ∪ B = {ω : ω ∈ A or ω ∈ B}

Contains all outcomes in A, all in B, and all common to both.

Event 'A and B' (Intersection)

**Definition:** A ∩ B represents "A and B simultaneously"

**Formula:**

A ∩ B = {ω : ω ∈ A and ω ∈ B}

Contains only outcomes common to both A and B.

**Example:** Two dice thrown, A = "first die shows 6", B = "sum ≥ 11"

A = {(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B = {(5,6), (6,5), (6,6)}

A ∩ B = {(6,5), (6,6)} represents "first die is 6 AND sum is at least 11"

Event 'A but not B' (Difference)

**Definition:** A – B represents "A occurs but B doesn't"

**Formula:**

A – B = A ∩ B′

Contains outcomes in A that are not in B.

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Mutually Exclusive Events

**Definition:** Two events A and B are **mutually exclusive** if they cannot occur simultaneously.

**Condition:** A ∩ B = φ (the sets are disjoint)

**Property:** Simple events of a sample space are always mutually exclusive.

**Example:** Rolling a die, A = "odd number appears" = {1, 3, 5}, B = "even number appears" = {2, 4, 6}

A ∩ B = φ, so A and B are mutually exclusive.

**Counterexample:** A = "odd number" = {1, 3, 5}, B = "number less than 4" = {1, 2, 3}

A ∩ B = {1, 3} ≠ φ, so they are NOT mutually exclusive.

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Exhaustive Events

**Definition:** Events E₁, E₂, ..., Eₙ are **exhaustive** if their union equals the sample space.

**Condition:**

E₁ ∪ E₂ ∪ ... ∪ Eₙ = S

This means at least one of the events must occur in every trial.

**Mutually Exclusive and Exhaustive Events:**

If events are:

1. Pairwise disjoint: Eᵢ ∩ Eⱼ = φ for i ≠ j

2. Their union is S: ⋃ᵢ₌₁ⁿ Eᵢ = S

Then they form a **partition** of S.

**Example:** Three coins tossed, A = "no heads" = {TTT}, B = "exactly one head" = {HTT, THT, TTH}, C = "at least two heads" = {HHT, HTH, THH, HHH}

A ∪ B ∪ C = S ✓ (exhaustive)

A ∩ B = φ, A ∩ C = φ, B ∩ C = φ ✓ (mutually exclusive)

These form a mutually exclusive and exhaustive set.

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Axiomatic Approach to Probability

**Definition:** Probability P is a real-valued function from the power set of S to [0, 1] satisfying three axioms:

**Axiom 1 (Non-negativity):** For any event E,

P(E) ≥ 0

**Axiom 2 (Certainty):** For the sample space S,

P(S) = 1

**Axiom 3 (Additivity):** If E and F are mutually exclusive events (E ∩ F = φ), then

P(E ∪ F) = P(E) + P(F)

**Consequence:** P(φ) = 0

Proof: Take any event E and F = φ. Since E ∩ φ = φ:

P(E ∪ φ) = P(E) + P(φ) → P(E) = P(E) + P(φ) → P(φ) = 0

Properties of Probability

**For a sample space S = {ω₁, ω₂, ..., ωₙ}:**

1. 0 ≤ P(ωᵢ) ≤ 1 for each elementary event ωᵢ

2. P(ω₁) + P(ω₂) + ... + P(ωₙ) = 1

3. For any event A ⊆ S: P(A) = Σ P(ωᵢ) for all ωᵢ ∈ A

**Multiple valid assignments exist:** For a coin toss, we can assign P(H) = ½, P(T) = ½ OR P(H) = ¼, P(T) = ¾, as long as both conditions are satisfied.

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Probability of an Event

**Definition:** The probability of an event A is the sum of probabilities of all elementary events in A.

**Formula:**

P(A) = Σ P(ωᵢ) for all ωᵢ ∈ A

**Example:** Examining three consecutive pens (B = bad, G = good)

S = {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG} with equal probabilities 1/8 each.

Event A: exactly one defective = {BGG, GBG, GGB}

P(A) = 1/8 + 1/8 + 1/8 = 3/8

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Equally Likely Outcomes

**Definition:** If all outcomes in a sample space are **equally likely**, each has the same probability.

**Formula:** For sample space with n outcomes:

P(ωᵢ) = 1/n for each ωᵢ

**Probability of an Event (Classical Definition):**

P(E) = Number of favorable outcomes / Total number of outcomes = n(E) / n(S)

**Key Condition:** This formula applies ONLY when all outcomes are equally likely.

**Example:** Rolling a fair die, E = "getting prime" = {2, 3, 5}

P(E) = 3/6 = 1/2

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Probability of 'A or B' (Addition Rule)

**Theorem:** For any two events A and B,

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

**Derivation:** A ∪ B can be partitioned as:

A ∪ B = (A – B) ∪ (A ∩ B) ∪ (B – A)

These three sets are mutually exclusive, so:

P(A ∪ B) = P(A – B) + P(A ∩ B) + P(B – A)

Also:

P(A) = P(A – B) + P(A ∩ B) → P(A – B) = P(A) – P(A ∩ B)

P(B) = P(B – A) + P(A ∩ B) → P(B – A) = P(B) – P(A ∩ B)

Therefore:

P(A ∪ B) = [P(A) – P(A ∩ B)] + P(A ∩ B) + [P(B) – P(A ∩ B)]

= P(A) + P(B) – P(A ∩ B)

**For Mutually Exclusive Events:** If A ∩ B = φ, then

P(A ∪ B) = P(A) + P(B)

**Example:** Tossing three coins, A = "at least 2 heads" = {HHT, HTH, THH, HHH}, B = "exactly 1 head" = {HTT, THT, TTH}

With equal probabilities:

P(A) = 4/8, P(B) = 3/8, A ∩ B = φ

P(A ∪ B) = 4/8 + 3/8 = 7/8

**Common Mistake:** Students often compute P(A) + P(B) without subtracting P(A ∩ B), which causes overlap.

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Probability of an Event and Its Complement

**Theorem:** For any event E and its complement E′,

P(E) + P(E′) = 1

Or equivalently:

P(E′) = 1 – P(E)

**Proof:** Since E and E′ partition S:

E ∪ E′ = S and E ∩ E′ = φ

By Axiom 3:

P(E ∪ E′) = P(E) + P(E′)

P(S) = P(E) + P(E′)

1 = P(E) + P(E′)

Therefore: P(E′) = 1 – P(E)

**Application:** If probability of rain is 0.4, probability of no rain is 0.6. This formula makes calculating "at least one" events easier—find probability of "none" and subtract from 1.

**Example:** Rolling a die, E = "getting less than 3" = {1, 2}

P(E) = 2/6 = 1/3

P(E′) = 1 – 1/3 = 2/3 (getting 3 or more)