Introduction to Probability

Probability or chance is a common term used in everyday life. For instance, we often say, "It may rain today", indicating uncertainty.

Probability is a quantitative measure of the likelihood or chance of occurrence of a particular event.

It is defined on a scale from 0 to 1, where:

0 means the event is impossible,
1 means the event is certain.

🔹 Random Experiment

An experiment is said to be a random experiment if:

All possible outcomes are known,
But the exact result cannot be predicted in advance.

📌 Examples of Random Experiments:

Tossing a fair coin → Outcomes: H or T
Throwing an unbiased die → Outcomes: 1 to 6
Drawing a card from a shuffled deck → Outcomes: Any one of the 52 cards
Picking a ball from a bag containing balls of different colors

🔹 Sample Space (S)

The sample space is the set of all possible outcomes of an experiment.

Experiment	Sample Space (S)
Tossing a coin	{H, T}
Rolling a die	{1, 2, 3, 4, 5, 6}
Tossing two coins	{HH, HT, TH, TT}
Drawing one card	52 possible outcomes

🔹 Equally Likely Events

Events are equally likely if each outcome has the same chance of occurring.

Examples:

Tossing a coin: H or T
Rolling a die: Any of 1 to 6

🔹 Classification of Events

Event Type	Definition	Example
Simple Event	Single outcome	Getting a '4' on a die
Compound Event	Combination of simple events	Getting an even number on a die
Mutually Exclusive	Events that cannot occur simultaneously	Getting H and T on a single coin toss
Not Mutually Exclusive	Events that can occur together	A = {2, 4, 6}, B = {4, 5, 6} ⇒ A ∩ B ≠ ϕ
Independent Events	Events where the outcome of one does not affect the other	Tossing a coin twice
Dependent Events	Events where the outcome of one affects the other	Drawing cards without replacement
Exhaustive Events	All possible outcomes of an experiment	Tossing 2 coins ⇒ 4 outcomes
Complementary Events	The event not occurring is called the complement (denoted by A̅)	If P(A) = 0.7 ⇒ P(A̅) = 0.3

🔹 Basic Probability Formula

P(E) = n(E) / n(S)

P(E): Probability of event E

n(E): Number of favorable outcomes

n(S): Total number of outcomes in sample space

Important Rules:

P(S) = 1
0 ≤ P(E) ≤ 1
P(ϕ) = 0

🔹 Addition Theorem

If A and B are two events:

P(A∪B) = P(A) + P(B) − P(A∩B)

If A and B are mutually exclusive:

P(A∪B) = P(A) + P(B)

🔹 Multiplication Theorem (Independent Events)

P(A∩B) = P(A) ⋅ P(B)

🔹 Complement Rule

P(A̅) = 1 - P(A)

🔹 Odds in Favor and Against

Odds in favor of event E: x : y → x/y

Odds against event E: y : x → y/x

🔹 Conditional Probability

The probability of an event given that another event has already occurred:

P(A∣B) = P(A∩B) / P(B), if P(B) > 0

Example:

In a box with 3 red and 2 blue balls, one ball is drawn and not replaced.

What is the probability the second ball is blue given the first was red?

P(B₂∣R₁) = (3/5 ⋅ 2/4) ÷ 3/5 = 1/2

🔹 Problems Based on Coins, Dice, and Cards

🪙 Coins:

1 coin ⇒ 2 outcomes: {H, T}
2 coins ⇒ 4 outcomes: {HH, HT, TH, TT}
n coins ⇒ 2ⁿ outcomes

🎲 Dice:

1 die ⇒ 6 outcomes: {1, 2, 3, 4, 5, 6}
2 dice ⇒ 36 outcomes
Example: Probability of sum = 7 ⇒ 6/36 = 1/6

🃏 Cards:

52 cards in a deck:
4 suits: Spades (♠), Clubs (♣), Hearts (♥), Diamonds (♦)
Each suit has: 13 cards (2–10, J, Q, K, A)
Face cards: J, Q, K ⇒ 12 total
Red cards: ♥, ♦ = 26
Black cards: ♠, ♣ = 26
Example: Probability of drawing an Ace = 4/52 = 1/13

🔹 Introduction to Probability