Probability Summary | Quantitative Aptitude for CA Foundation PDF Download

Introduction

• The words like chance, probable, likely, odds come from the study of probability.
• Probability is now an important branch of statistics and is used in hypothesis testing, estimation, forecasting, and decision-making.
• The first use of probability was more than 300 years ago, mainly in gambling problems by European mathematicians.
• Later, famous mathematicians like De Moivre, Laplace, Bayes, Fisher, Chebyshev, Markov, Kolmogorov developed the full theory.
• Probability helps us deal with uncertainty and understand how likely an event is to happen.
• The chapter focuses on Objective Probability, not subjective (personal judgment–based) probability.

Random Experiment

• An experiment is any action that produces a result.
• A random experiment is one where the outcome cannot be predicted with certainty.
• Even if the experiment is repeated under the same conditions, the exact outcome is not known in advance.
• Examples of random experiments from the chapter:
  1. Tossing a coin → result can be Head or Tail
  2. Rolling a die → any number from 1 to 6 may appear
  3. Drawing cards from a shuffled pack
  4. Picking defective or non-defective items from a box

Event

• An outcome or set of outcomes of a random experiment.
• Simple Event: Single outcome.
• Compound Event: Multiple outcomes.
• Mutually Exclusive: Cannot happen together.
• Exhaustive: All possible outcomes.
• Equally Likely: Same chance of occurring.

Random Variable

• A variable that assigns numbers to outcomes of a random experiment.
• Discrete: Countable values (0,1,2…).
• Continuous: Infinite values (height, weight).
• Has a Probability Distribution listing each value with its probability.

Defective Items

• Problems where a sample is chosen from items containing defective (D) and non-defective (D′) items.
• Uses combinations to count how many sample selections include 0, 1, or more defectives.
• General formula used:
• Example: From 10 items (3 defective), a probability sample of 4 has not more than 1 defective = 2/3.

Classical Probability

• Classical probability applies when all outcomes are finite and equally likely.
• It is also called the A Priori Definition (based on prior knowledge of all possible outcomes).
• Formula: 

• Used in situations like coins, dice, and cards—where outcomes are known in advance.
• Probability always lies between 0 and 1.
• If P(A) = 0, event A is impossible.
• If P(A) = 1, event A is certain.

Relative Frequency Definition of Probability

• This definition is also called the Statistical Definition of Probability.
• It is used when outcomes are not known in advance or are not equally likely.
• Based on actual observations from repeated experiments.

Definition: If an experiment is repeated n times under identical conditions and an event A occurs fₐ times, then:

• Means: Probability = Long-run proportion of times the event happens.
• As the number of trials increases, the ratio becomes more stable.
• Useful in real-life uncertain situations:
  Weather forecasting
  Quality control
  Mortality/life tables
  Insurance
• Does not require equally likely outcomes.

Key Points

• Based on experimental data, not theory.
• Gives estimated probability, not exact.
• More realistic than the classical definition in many situations.

Example: Wage distribution of 150 workers:

Number earning less than ₹80 = 74

Operations on Events – Set Theoretic Approach

Sample space (S) is the set of all possible outcomes.
An event (A) is any subset of S.

Operations on events:

• Union (A ∪ B): A occurs or B occurs or both.
• Intersection (A ∩ B): A and B occur together.
• Difference (A – B): A occurs but B does not.
• Complement (A′): A does not occur (S – A).

Types of events:

• Mutually exclusive: cannot happen together (A ∩ B = Ø).
• Exhaustive: together cover all outcomes (A ∪ B = S).
• Equally likely: have the same chance.

Union and Intersection of Two Events

Union (A ∪ B) includes all outcomes in A or B or both.
Intersection (A ∩ B) includes only outcomes common to both.

Probability formulas:

• For mutually exclusive events:
  P(A ∪ B) = P(A) + P(B)
• For any two events:
  P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
• Only A: P(A – B) = P(A) – P(A ∩ B)
• Only B: P(B – A) = P(B) – P(A ∩ B)

Axiomatic or Modern Definition of Probability

Probability satisfies three axioms:

1. P(A) ≥ 0
2. P(S) = 1
3. For mutually exclusive events A₁, A₂, A₃…
   P(A₁ ∪ A₂ ∪ A₃ ...) = P(A₁) + P(A₂) + P(A₃) + …

Useful results:

• P(A′) = 1 – P(A)
• 0 ≤ P(A) ≤ 1

Conditional Probability and Compound Theorem of Probability

• Conditional probability:
  P(B|A) = P(A ∩ B) / P(A)
• Independent events:
  P(A ∩ B) = P(A) × P(B)

Compound probability:

• For two events:
  P(A ∩ B) = P(A) × P(B|A)
• For three events:
  P(A ∩ B ∩ C) = P(A) × P(B|A) × P(C|A ∩ B)

Random Variable – Probability Distribution

• A random variable assigns numbers to outcomes.
• Types:
  - Discrete random variable: takes countable values.
  - Continuous random variable: takes any value in an interval.
• Probability distribution lists values of the random variable with their probabilities.
• Conditions:
  P(X = xᵢ) ≥ 0
• Sum of all probabilities = 1
• PMF (for discrete X): f(x) = P(X = x)
  PDF (for continuous X): f(x) ≥ 0 and integral of f(x) over its range = 1

Expected Value of a Random Variable

• Expected value (mean):
  E(X) = Σ xᵢ P(xᵢ)
• Expected value of a function g(X):
  E[g(X)] = Σ g(xᵢ) P(xᵢ)
• Variance:
  σ² = E(X²) – [E(X)]²
• Standard deviation:
  σ = √σ²
• If Y = a + bX:
• E(Y) = a + bE(X)
  σᵧ = |b| σₓ