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🎲 PROBABILITY: A DEEP MATHEMATICAL OVERVIEW
1. What Is Probability?

Probability is the mathematical study of randomness and uncertainty. It provides a framework to quantify the likelihood of events occurring in a well-defined experiment.

Formal Definition:
A probability is a function P that assigns a number between 0 and 1 to each event E in a sample space Ω, satisfying:

Non-negativity: P(E) ≥ 0
Normalization: P(Ω) = 1
Additivity: If A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B)




2. Foundational Terminology

Experiment: A process with uncertain outcomes (e.g., rolling a die).
Sample Space (Ω): Set of all possible outcomes.
Event (E): A subset of the sample space.
Elementary Event: A single outcome in Ω.
Mutually Exclusive Events: Events that cannot occur simultaneously.
Exhaustive Events: A set of events covering the entire sample space.


3. Types of Probability

Classical (Theoretical):
Based on symmetry and equally likely outcomes.
P(E) = Number of favorable outcomes / Total outcomes
Empirical (Experimental):
Based on observed data.
P(E) ≈ Frequency of E / Total trials
Subjective:
Based on personal belief or estimation.
Axiomatic (Kolmogorov):
Based on a formal system of axioms, allowing for complex and abstract probability spaces.


4. Key Probability Rules

Complement Rule:
P(E') = 1 − P(E)
Addition Rule:

For mutually exclusive events:
P(A ∪ B) = P(A) + P(B)
For general events:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)


Multiplication Rule:

For independent events:
P(A ∩ B) = P(A) × P(B)
For dependent events:
P(A ∩ B) = P(A) × P(B | A)


Conditional Probability:
P(A | B) = P(A ∩ B) / P(B), provided P(B) ≠ 0


5. Bayes’ Theorem

Used to update probabilities based on new evidence:

P(A | B) = [P(B | A) × P(A)] / P(B)

This is foundational in fields like machine learning, medical diagnosis, and decision theory.

6. Random Variables

A random variable (X) is a function that assigns a real number to each outcome in a sample space.

Discrete Random Variable: Takes countable values (e.g., number of heads in 3 coin tosses).
Continuous Random Variable: Takes values from a continuum (e.g., height, weight).


7. Probability Distributions

Discrete:

Probability Mass Function (PMF): P(X = x)
Examples: Binomial, Poisson, Geometric


Continuous:

Probability Density Function (PDF): f(x), where P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx
Examples: Normal, Exponential, Uniform


Cumulative Distribution Function (CDF):
F(x) = P(X ≤ x), applicable to both discrete and continuous variables.


8. Expected Value and Variance

Expected Value (Mean):

Discrete: E[X] = Σ x·P(X = x)
Continuous: E[X] = ∫ x·f(x) dx


Variance:
Var(X) = E[(X − E[X])²] = E[X²] − (E[X])²
Standard Deviation:
SD(X) = √Var(X)


9. Law of Large Numbers (LLN)

As the number of trials increases, the sample mean converges to the expected value. This justifies empirical probability.

10. Central Limit Theorem (CLT)

For large samples, the distribution of the sample mean approaches a normal distribution, regardless of the original distribution. This is foundational in statistics and hypothesis testing.

The questions are going to be posted @ 3

📘 Multiple Choice Questions on Probability

1. What is the probability of getting a prime number when a die is rolled once?
A) 1/2 B) 1/3 C) 2/3 D) 5/6

2. If two coins are tossed, what is the probability of getting at least one head?
A) 1/4 B) 1/2 C) 3/4 D) 1

3. Which of the following is a valid probability value?
A) -0.2 B) 0.5 C) 1.2 D) 2

4. The sum of probabilities of all elementary events of an experiment is:
A) 0 B) 1 C) Depends on the experiment D) Infinity

5. If A and B are independent events, then P(A ∩ B) equals:
A) P(A) + P(B) B) P(A) × P(B) C) P(A) − P(B) D) P(A | B)

6. In a deck of 52 cards, what is the probability of drawing a king or a queen?
A) 1/13 B) 2/13 C) 1/26 D) 1/52

7. Which of the following is not a type of probability?
A) Classical B) Empirical C) Subjective D) Objective

8. Bayes’ Theorem is used to calculate:
A) Joint probability B) Marginal probability C) Conditional probability D) Independent events

9. If P(A) = 0.7 and P(B) = 0.5, what is the maximum value of P(A ∩ B)?
A) 0.2 B) 0.5 C) 0.7 D) 1.2

10. The probability of an impossible event is:
A) 0 B) 1 C) 0.5 D) Undefined

11. Which distribution is used to model the number of successes in a fixed number of independent trials?
A) Poisson B) Binomial C) Normal D) Exponential

12. What is the expected value of a fair six-sided die roll?
A) 3 B) 3.5 C) 4 D) 4.5

13. If events A and B are mutually exclusive, then P(A ∩ B) is:
A) 0 B) 1 C) P(A) + P(B) D) P(A) × P(B)

14. Which of the following is a continuous probability distribution?
A) Binomial B) Poisson C) Normal D) Geometric

15. The variance of a fair coin toss (1 for heads, 0 for tails) is:
A) 0.25 B) 0.5 C) 1 D) 2

16. If P(A ∪ B) = 0.9, P(A) = 0.6, and P(B) = 0.5, then P(A ∩ B) is:
A) 0.2 B) 0.1 C) 0.0 D) 0.3

17. Which of the following statements is true for independent events A and B?
A) P(A | B) = P(A) B) P(A ∩ B) = 0 C) P(A ∪ B) = 1 D) P(B | A) = 0

18. In a probability distribution, the sum of all probabilities must be:
A) 0 B) 1 C) Less than 1 D) Greater than 1

19. The Central Limit Theorem applies when:
A) The sample size is small
B) The population is normal
C) The sample size is large
D) The variance is unknown

20. Which of the following is not a property of probability?
A) P(E) ≥ 0
B) P(Ω) = 1
C) P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
D) P(A ∩ B) = P(A) + P(B)