Unit–III: Probability and Probability Distributions

1. Introduction to Probability & Basic Concepts

Probability is a quantitative measure of uncertainty, expressing the likelihood that a specific event will occur on a scale from 0 (impossible event) to 1 (certain event).

PROBABILITY ARCHITECTURE
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RANDOM EXPERIMENT SAMPLE SPACE (S)
An operation or trial that yields unpredictable outcomes The set of all possible outcomes of a random
(e.g., tossing a coin, rolling a die). experiment.
 

Fundamental Terminology

  • Event (E): A subset of the sample space S representing a specific outcome or combination of outcomes.
  • Mutually Exclusive Events: Events that cannot occur simultaneously (A ∩ B = ∅). If event A happens, event B cannot happen.
  • Independent Events: Events where the occurrence or non-occurrence of one does not affect the probability of the other.
  • Exhaustive Events: A set of events whose union forms the entire sample space S, covering all possible outcomes.

2. Classical Definition of Probability

The Classical (or Prior) Definition assumes that all outcomes of a random experiment are equally likely and mutually exclusive.

P(A) = m / n = (Number of favorable outcomes for Event A) / (Total number of possible outcomes in Sample Space S)

Axioms of Probability

  • 0 ≤ P(A) ≤ 1
  • P(S) = 1
  • P(Ā) = 1 − P(A) (where Ā is the complement of A)

3. Probability Theorems: Addition & Multiplication Rules

PROBABILITY THEOREMS
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ADDITION RULES (OR / Union) MULTIPLICATION RULES (AND / Intersection)
Used when determining P(A ∪ B). Used when determining P(A ∩ B).
 
Scenario Operational Formula Key Condition
Addition Rule (General) P(A ∪ B) = P(A) + P(B) − P(A ∩ B) General theorem for any two events.
Addition Rule (Mutually Exclusive) P(A ∪ B) = P(A) + P(B) For Mutually Exclusive Events where P(A ∩ B) = 0.
Multiplication Rule (General) P(A ∩ B) = P(A) · P(B | A) General theorem using conditional probability.
Multiplication Rule (Independent) P(A ∩ B) = P(A) · P(B) For Independent Events where P(B | A) = P(B).

4. Probability Distributions: Binomial, Poisson, and Normal

A Probability Distribution is a mathematical function that models the likelihood of different outcomes in an experiment.

PROBABILITY DISTRIBUTIONS
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DISCRETE DISTRIBUTIONS CONTINUOUS DISTRIBUTIONS
Takes countable integer values (x = 0, 1, 2, …) Takes any continuous real value within a range
• Binomial Distribution • Normal Distribution (Gaussian)
• Poisson Distribution
 

1. Binomial Distribution

A discrete distribution modeling the number of successes in n independent Bernoulli trials, where each trial has only two outcomes (Success p, Failure q = 1 − p).

P(X = k) = C(n, k) pk qn−k = [n! / (k!(n−k)!)] pk qn−k
  • Mean: μ = np
  • Variance: σ² = npq
  • Standard Deviation: σ = √(npq)

2. Poisson Distribution

A discrete distribution that models the count of rare events occurring randomly over a fixed interval of time or space. It acts as a limiting form of the Binomial distribution when n → ∞ and p → 0, such that λ = np remains constant.

P(X = k) = e−λ λk / k!
  • Mean: μ = λ
  • Variance: σ² = λ (Mean and Variance are equal)

3. Normal Distribution (Gaussian Curve)

A continuous, symmetrical, bell-shaped probability distribution defined by its mean (μ) and standard deviation (σ).

f(x) = [1 / (σ√(2π))] · e−½((x−μ)/σ)²

NORMAL CURVE SHAPE
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Area under the entire curve = 1.0
Symmetrical about Mean (μ)
Mean = Median = Mode
 
Standard Normal Variable (Z) = (X − μ) / σ

Empirical Rule Boundaries

  • μ ± 1σ covers 68.27% of total observations.
  • μ ± 2σ covers 95.45% of total observations.
  • μ ± 3σ covers 99.73% of total observations.

5. Expected Value E(X)

The Expected Value represents the long-run average or weighted mean outcome of a random variable.

Variable Type Expected Value Formula E(X) Variance Formula Var(X)
Discrete Variable E(X) = Σ x · P(x) Var(X) = E(X²) − [E(X)]²
Continuous Variable E(X) = ∫−∞∞ x · f(x) dx Var(X) = ∫−∞∞ x² f(x) dx − μ²

Algebraic Properties of Expected Value

  • E(c) = c (Expected value of a constant is the constant itself)
  • E(aX + b) = aE(X) + b
  • E(X + Y) = E(X) + E(Y)
  • E(XY) = E(X) · E(Y) (if variables X and Y are independent)

Key Learning Summary

Topic Core Operational Takeaway
Classical Probability Calculated as P(A) = Favorable / Total outcomes.
Addition Theorem P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Multiplication Theorem P(A ∩ B) = P(A) · P(B) for independent events.
Binomial vs. Poisson Binomial has fixed n and p; Poisson models rare events with Mean = Variance = λ.
Normal Distribution Bell-shaped continuous distribution centered at μ, analyzed using standard scores (Z).
Expected Value Weighted average defined as E(X) = Σ x · P(x).

 

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