Probability Theory Basics – A Comprehensive Guide

Probability theory is the branch of mathematics that deals with quantifying uncertainty. It is widely used in machine learning, artificial intelligence, finance, statistics, data science, physics, and everyday decision-making.

In this detailed guide, we will cover:

1. Introduction to Probability
2. Basic Probability Concepts
3. Types of Probability
4. Probability Rules
5. Conditional Probability & Bayes’ Theorem
6. Random Variables & Probability Distributions
7. Expectation & Variance
8. Law of Large Numbers & Central Limit Theorem
9. Applications in Machine Learning

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1. Introduction to Probability

1.1 What is Probability?

Probability measures the likelihood of an event occurring. It is defined mathematically as: P(E)=Number of favorable outcomesTotal number of possible outcomesP(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

where:

• P(E)P(E) is the probability of event EE.
• The probability value lies between 0 and 1:
  • P(E)=0P(E) = 0 (Impossible event)
  • P(E)=1P(E) = 1 (Certain event)

1.2 Examples of Probability

• Tossing a fair coin: P(Heads)=12P(\text{Heads}) = \frac{1}{2}
• Rolling a die: P(Rolling a 4)=16P(\text{Rolling a 4}) = \frac{1}{6}
• Drawing a red card from a standard deck of 52 cards: P(Red)=2652=12P(\text{Red}) = \frac{26}{52} = \frac{1}{2}

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2. Basic Probability Concepts

2.1 Experiment

An experiment is an action or process that leads to an outcome (e.g., rolling a die).

2.2 Sample Space (SS)

The sample space is the set of all possible outcomes.

• Example: Tossing a coin → S={H,T}S = \{H, T\}
• Example: Rolling a die → S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}

2.3 Event (EE)

An event is a subset of the sample space.

• Example: Rolling an even number → E={2,4,6}E = \{2, 4, 6\}

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3. Types of Probability

3.1 Classical Probability

When all outcomes are equally likely: P(E)=Favorable outcomesTotal outcomesP(E) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}

Example: Probability of drawing an Ace from a deck: P(Ace)=452=113P(Ace) = \frac{4}{52} = \frac{1}{13}

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3.2 Empirical (Experimental) Probability

Based on actual experiments or past data: P(E)=Number of times event occursTotal number of trialsP(E) = \frac{\text{Number of times event occurs}}{\text{Total number of trials}}

Example: If a coin is flipped 100 times and lands on heads 48 times: P(Heads)=48100=0.48P(Heads) = \frac{48}{100} = 0.48

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3.3 Subjective Probability

Based on intuition, personal belief, or experience.
Example: A weather forecast predicting a 70% chance of rain.

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4. Probability Rules

4.1 Addition Rule

For two mutually exclusive events AA and BB: P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)

Example: Rolling a 1 or 6 on a fair die: P(1 or 6)=P(1)+P(6)=16+16=26=13P(1 \text{ or } 6) = P(1) + P(6) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}

For non-mutually exclusive events: P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) – P(A \cap B)

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4.2 Multiplication Rule

For two independent events: P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)

Example: Probability of getting heads twice in a row: P(HH)=P(H)×P(H)=12×12=14P(HH) = P(H) \times P(H) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

For dependent events: P(A∩B)=P(A)×P(B∣A)P(A \cap B) = P(A) \times P(B|A)

where P(B∣A)P(B|A) is the conditional probability of BB given AA.

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5. Conditional Probability & Bayes’ Theorem

5.1 Conditional Probability

The probability of event BB occurring given that AA has already occurred: P(B∣A)=P(A∩B)P(A)P(B|A) = \frac{P(A \cap B)}{P(A)}

Example: Probability of drawing two aces in a row from a deck: P(A1)=452,P(A2∣A1)=351P(A_1) = \frac{4}{52}, \quad P(A_2 | A_1) = \frac{3}{51} P(A1∩A2)=452×351=122652=1221P(A_1 \cap A_2) = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}

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5.2 Bayes’ Theorem

Bayes’ theorem helps update probabilities with new evidence: P(A∣B)=P(B∣A)P(A)P(B)P(A|B) = \frac{P(B|A) P(A)}{P(B)}

Example: Used in spam filtering, medical diagnosis, and machine learning.

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6. Random Variables & Probability Distributions

6.1 Random Variables

A random variable assigns numerical values to outcomes.

• Discrete Random Variable: Takes countable values (e.g., rolling a die).
• Continuous Random Variable: Takes infinite values (e.g., height, weight).

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6.2 Probability Distributions

A probability distribution shows how probabilities are assigned to different outcomes.

• Discrete Probability Distribution: Example: Binomial Distribution
• Continuous Probability Distribution: Example: Normal (Gaussian) Distribution

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7. Expectation & Variance

7.1 Expected Value (E(X)E(X))

The mean of a probability distribution: E(X)=∑xP(x)E(X) = \sum x P(x)

7.2 Variance (Var(X)Var(X))

Measures how spread out values are: Var(X)=E(X2)−(E(X))2Var(X) = E(X^2) – (E(X))^2

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8. Law of Large Numbers & Central Limit Theorem

8.1 Law of Large Numbers (LLN)

As the number of trials increases, the empirical probability approaches the theoretical probability.

8.2 Central Limit Theorem (CLT)

The sum of a large number of independent random variables follows a normal distribution.

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9. Applications of Probability in Machine Learning

• Naïve Bayes Classifier: Uses Bayes’ theorem for classification.
• Markov Chains: Models probability transitions between states.
• Hidden Markov Models (HMMs): Used in speech recognition.
• Gaussian Mixture Models (GMMs): Used in clustering.

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10. Summary Table

Concept	Formula	Example
Probability	P(E)=favorable outcomestotal outcomesP(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}	P(Heads)=12P(\text{Heads}) = \frac{1}{2}
Addition Rule	P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) – P(A \cap B)	Rolling 1 or 2
Multiplication Rule	P(A∩B)=P(A)P(B)P(A \cap B) = P(A) P(B)	Getting heads twice
Bayes’ Theorem	( P(A	B) = \frac{P(B

Would you like Python code examples for these concepts?