Introduction
Basic Terminology
Types of Events
Probability Rules
Conditional Probability
Law of Total Probability
Bayes’ Theorem
Random Variable
Mean, Median, Mode
Variance & Standard Deviation
Probability Distributions
Bernoulli Distribution
Binomial Distribution
Uniform Distribution
Normal Distribution
Poisson Distribution
AI/ML Applications
Formula Cheat Sheet

1. Introduction

Probability is a branch of mathematics that measures the likelihood of an event occurring. It answers questions like:

What is the chance of getting a Head?
What is the probability a customer buys a product?
What is the probability an email is spam?

In Machine Learning, probability helps in:

Classification
Prediction
Decision Making
Uncertainty Estimation
Bayesian Learning

Probability values lie between

0 \le P(E) \le 1

where

0 → Impossible event
1 → Certain event

2. Basic Terminology

Experiment

An action that produces outcomes. Examples:

Tossing a coin
Rolling a die
Picking a card

Sample Space (S)

Set of all possible outcomes. Coin

S={H,T}

Die

S={1,2,3,4,5,6}

Event (E)

A subset of the sample space. Example Rolling an even number

E={2,4,6}

Probability of an Event

P(E)=\frac{\text{Number of favorable outcomes}} {\text{Total outcomes}}

Example 1: Coin Toss

Sample Space

{H,T}

Event Getting Head

P(H)=\frac12

sample = ["H", "T"]
p_head = 1 / len(sample)
print(p_head)

Output

0.5

Example 2: Roll a Die

Event Even number

E={2,4,6}

P(E)=\frac36=\frac12

sample = [1,2,3,4,5,6]
even = [2,4,6]

print(len(even)/len(sample))

Example 3: Coin + Die

Event Head AND Even Number Total outcomes

2\times6=12

Favorable

(H,2),(H,4),(H,6)

P(E)=\frac3{12}=\frac14

3. Types of Events

Simple Event

Contains one outcome. Example Getting 4 on a die.

Compound Event

Contains multiple outcomes. Example Getting an even number.

Certain Event

Always occurs.

P(E)=1

Example Rolling a number less than 7.

Impossible Event

Never occurs.

P(E)=0

Example Rolling 8 on a standard die.

Union of Events

“A or B”

A\cup B

Example Even OR Odd

Intersection of Events

“A and B”

A\cap B

Example Even AND Multiple of 3 Result

{6}

Mutually Exclusive Events

Cannot occur together.

A\cap B=\varnothing

Example Head and Tail in one coin toss.

Independent Events

One event does not affect another. Example Two coin tosses.

Dependent Events

One event affects another. Example Drawing cards without replacement.

4. Probability Rules

Rule 1: Complement Rule

Probability that an event does not occur.

P(A')=1-P(A)

Example

P(\text{Not Head})=1-\frac12=\frac12

Addition Rule

Mutually Exclusive

P(A\cup B)=P(A)+P(B)

Example Roll a die A = 2 B = 4

P(A)=\frac16

P(B)=\frac16

P(A\cup B)=\frac16+\frac16=\frac13

Non-Mutually Exclusive

P(A\cup B)=P(A)+P(B)-P(A\cap B)

Example A = Even B = Multiple of 3

A={2,4,6}

B={3,6}

Intersection

{6}

P(A)=\frac36

P(B)=\frac26

P(A\cap B)=\frac16

P(A\cup B)=\frac36+\frac26-\frac16=\frac46=\frac23

Multiplication Rule

Independent

P(A\cap B)=P(A)P(B)

Example Two coins

P(HH)=\frac12\times\frac12=\frac14

Dependent

P(A\cap B)=P(A)P(B|A)

5. Conditional Probability

Probability of B given A occurred.

P(B|A)= \frac{P(A\cap B)} {P(A)}

Example A = Even B = Multiple of 3

P(B|A)=\frac{1/6}{3/6} =\frac13

p_intersection = 1/6
p_even = 3/6

print(p_intersection/p_even)

6. Law of Total Probability

A_1,A_2,...A_n

partition the sample space, then

P(B)= \sum_i P(B|A_i)P(A_i)

Used when probability comes from multiple possible cases. Example Medical diagnosis Different age groups contribute differently to disease probability.

7. Bayes’ Theorem

Used to update probabilities after observing evidence.

P(A|B)= \frac{P(B|A)P(A)} {P(B)}

Where

(P(A)) = Prior
(P(B|A)) = Likelihood
(P(B)) = Evidence
(P(A|B)) = Posterior

Applications

Spam filtering
Medical diagnosis
Recommendation systems
Naive Bayes Classifier

Example Disease prevalence

P(D)=0.01

Test accuracy

P(+|D)=0.99

False positive

P(+|\bar D)=0.05

Use Bayes’ theorem to compute

P(D|+)

8. Random Variable

A variable whose value depends on a random experiment.

Discrete Random Variable

Countable values. Example Number of Heads.

Continuous Random Variable

Infinite values. Example Height Weight Temperature

9. Mean, Median, Mode

Mean

Average value.

\mu= \frac{\sum x}{n}

Median

Middle value after sorting.

Mode

Most frequently occurring value.

10. Variance & Standard Deviation

Variance

Measures spread. Population

\sigma^2= \frac{\sum(x-\mu)^2} N

Sample

s^2= \frac{\sum(x-\bar x)^2} {n-1}

Standard Deviation

Square root of variance.

\sigma=\sqrt{\sigma^2}

Higher SD ⇒ More spread. Lower SD ⇒ Less spread.

import numpy as np

data = [2,4,6,8]

print(np.mean(data))
print(np.var(data))
print(np.std(data))

11. Probability Distribution

Describes how probabilities are assigned to values. Two types

Discrete Distribution
Continuous Distribution

Common distributions

Bernoulli
Binomial
Uniform
Normal
Poisson

12. Bernoulli Distribution

Single trial. Only two outcomes.

Success
Failure

Parameter

p

Examples

Pass/Fail
Spam/Not Spam
Click/No Click

Mean

p

Variance

p(1-p)

13. Binomial Distribution

Extension of Bernoulli. Conditions

Fixed number of trials
Independent trials
Same probability

Formula

P(X=k)= \binom nk p^k (1-p)^{n-k}

Example Probability of exactly 3 Heads in 5 tosses. Applications

Customer conversions
Defect detection

14. Uniform Distribution

Every value is equally likely. PDF

f(x)= \frac1{b-a}

Example Random number between 0 and 1. Applications

Random initialization
Simulation

15. Normal Distribution

Most important distribution in ML. Bell-shaped curve. Properties

Symmetric
Mean = Median = Mode
Area = 1

PDF

f(x)= \frac1{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

68-95-99.7 Rule

68% within 1 SD
95% within 2 SD
99.7% within 3 SD

Applications

Gaussian Naive Bayes
Noise Modeling
Statistics
Data Normalization

16. Poisson Distribution

Models the number of events occurring in a fixed interval. Formula

P(X=k)= \frac{e^{-\lambda}\lambda^k} {k!}

Applications

Calls per minute
Website hits
Machine failures
Customer arrivals

17. AI/ML Applications

Topic	AI/ML Application
Probability	Classification
Conditional Probability	Bayesian Networks
Bayes’ Theorem	Naive Bayes
Mean	Feature Scaling
Variance	Data Spread
Standard Deviation	Outlier Detection
Bernoulli	Binary Classification
Binomial	Conversion Prediction
Normal	Gaussian Models
Poisson	Event Prediction

18. Formula Cheat Sheet

Probability

P(E)=\frac{\text{Favorable}}{\text{Total}}

Complement

P(A')=1-P(A)

Addition Rule

P(A\cup B)=P(A)+P(B)-P(A\cap B)

Multiplication Rule

P(A\cap B)=P(A)P(B)

Conditional Probability

P(B|A)= \frac{P(A\cap B)} {P(A)}

Total Probability

P(B)=\sum P(B|A_i)P(A_i)

Bayes’ Theorem

P(A|B)= \frac{P(B|A)P(A)} {P(B)}

Mean

\mu=\frac{\sum x}{n}

Variance

\sigma^2=\frac{\sum(x-\mu)^2}{N}

Standard Deviation

\sigma=\sqrt{\sigma^2}

Binomial Distribution

P(X=k)= \binom nk p^k (1-p)^{n-k}

Uniform Distribution

f(x)=\frac1{b-a}

Normal Distribution

f(x)= \frac1{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Poisson Distribution

P(X=k)= \frac{e^{-\lambda}\lambda^k} {k!}

Key Takeaways

Probability measures uncertainty and ranges from 0 to 1.
Understand sample space, events, and probability rules before advanced topics.
Conditional Probability, Law of Total Probability, and Bayes’ Theorem are foundational for probabilistic ML.
Random variables describe outcomes numerically.
Mean, variance, and standard deviation summarize data.
The most important probability distributions for AI/ML are Bernoulli, Binomial, Uniform, Normal, and Poisson.
These concepts underpin algorithms such as Naive Bayes, Gaussian Models, Bayesian Networks, and many statistical learning methods.

​1. Introduction

​2. Basic Terminology

​Experiment

​Sample Space (S)

​Event (E)

​Probability of an Event

​Example 1: Coin Toss

​Example 2: Roll a Die

​Example 3: Coin + Die

​3. Types of Events

​Simple Event

​Compound Event

​Certain Event

​Impossible Event

​Union of Events

​Intersection of Events

​Mutually Exclusive Events

​Independent Events

​Dependent Events

​4. Probability Rules

​Rule 1: Complement Rule

​Addition Rule

​Mutually Exclusive

​Non-Mutually Exclusive

​Multiplication Rule

​Independent

​Dependent

​5. Conditional Probability

​6. Law of Total Probability

​7. Bayes’ Theorem

​8. Random Variable

​Discrete Random Variable

​Continuous Random Variable

​9. Mean, Median, Mode

​Mean

​Median

​Mode

​10. Variance & Standard Deviation

​Variance

​Standard Deviation

​11. Probability Distribution

​12. Bernoulli Distribution

​13. Binomial Distribution

​14. Uniform Distribution

​15. Normal Distribution

​16. Poisson Distribution

​17. AI/ML Applications

​18. Formula Cheat Sheet

​Probability

​Complement

​Addition Rule

​Multiplication Rule

​Conditional Probability

​Total Probability

​Bayes’ Theorem

​Mean

​Variance

​Standard Deviation

​Binomial Distribution

​Uniform Distribution

​Normal Distribution

​Poisson Distribution

​Key Takeaways

1. Introduction

2. Basic Terminology

Experiment

Sample Space (S)

Event (E)

Probability of an Event

Example 1: Coin Toss

Example 2: Roll a Die

Example 3: Coin + Die

3. Types of Events

Simple Event

Compound Event

Certain Event

Impossible Event

Union of Events

Intersection of Events

Mutually Exclusive Events

Independent Events

Dependent Events

4. Probability Rules

Rule 1: Complement Rule

Addition Rule

Mutually Exclusive

Non-Mutually Exclusive

Multiplication Rule

Independent

Dependent

5. Conditional Probability

6. Law of Total Probability

7. Bayes’ Theorem

8. Random Variable

Discrete Random Variable

Continuous Random Variable

9. Mean, Median, Mode

Mean

Median

Mode

10. Variance & Standard Deviation

Variance

Standard Deviation

11. Probability Distribution

12. Bernoulli Distribution

13. Binomial Distribution

14. Uniform Distribution

15. Normal Distribution

16. Poisson Distribution

17. AI/ML Applications

18. Formula Cheat Sheet

Probability

Complement

Addition Rule

Multiplication Rule

Conditional Probability

Total Probability

Bayes’ Theorem

Mean

Variance

Standard Deviation

Binomial Distribution

Uniform Distribution

Normal Distribution

Poisson Distribution

Key Takeaways