Sample Spaces, Events & Probability

Understanding the fundamentals of probability theory

What is Probability?

Probability is the mathematical measure of the likelihood of an event occurring. It quantifies uncertainty and helps us predict the outcomes of random experiments. Probability values range from 0 (impossible) to 1 (certain).

Random Experiments & Sample Spaces

Random Experiment

An experiment whose outcome cannot be predicted with certainty is called a random experiment. The outcomes must be:

Well-defined: All possible outcomes are known
Unpredictable: The exact outcome in any single trial cannot be determined
Repeatable: The experiment can be performed any number of times

Examples: Tossing a coin, rolling a die, drawing cards from a deck, spinning a wheel

Sample Space (S)

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

Example 1: Sample Space of a Coin Toss

Experiment: Toss a fair coin once

Sample Space: S = {H, T}

Number of outcomes: n(S) = 2

Example 2: Sample Space of Rolling a Die

Experiment: Roll a fair six-sided die

Sample Space: S = {1, 2, 3, 4, 5, 6}

Number of outcomes: n(S) = 6

Example 3: Sample Space of Two Coins

Experiment: Toss two fair coins simultaneously

Sample Space: S = {HH, HT, TH, TT}

Number of outcomes: n(S) = 4

Events

An event is a subset of the sample space. It can be:

Types of Events

📌 Simple Event (Elementary Event)

An event containing only one outcome from the sample space.

Example: In rolling a die, the event "getting 3" = {3}

📌 Compound Event

An event containing more than one outcome from the sample space.

Example: In rolling a die, the event "getting an even number" = {2, 4, 6}

📌 Sure Event (Certain Event)

An event that is certain to occur. It equals the sample space S.

Example: In rolling a die, "getting a number from 1 to 6" = S

📌 Impossible Event

An event that cannot occur. It is represented by the empty set ∅.

Example: In rolling a die, "getting 7" = ∅

Complementary Event

If E is an event, then E' (or E^c) is the complementary event containing all outcomes in S but not in E.

Example: Complementary Event

Sample Space: S = {1, 2, 3, 4, 5, 6}

Event E: Getting an odd number = {1, 3, 5}

Complementary Event E': Not getting an odd number = {2, 4, 6}

Probability of an Event

Classical (Theoretical) Probability

For a random experiment with n equally likely outcomes:

Probability Formula

P(E) = Number of favorable outcomes / Total number of outcomes = n(E) / n(S)

Where E is the event and S is the sample space

📌 Properties of Probability

0 ≤ P(E) ≤ 1 for any event E
P(S) = 1 (probability of sure event)
P(∅) = 0 (probability of impossible event)
P(E) + P(E') = 1, so P(E') = 1 - P(E)

Worked Examples

Example 1: Probability from a Die

Problem: A fair die is rolled. Find the probability of getting (a) an even number (b) a number greater than 4

Solution:

Sample Space: S = {1, 2, 3, 4, 5, 6}, n(S) = 6

(a) Even number: E = {2, 4, 6}, n(E) = 3

P(E) = 3/6 = 1/2 = 0.5

(b) Number greater than 4: E = {5, 6}, n(E) = 2

P(E) = 2/6 = 1/3 ≈ 0.333

Example 2: Probability from a Deck of Cards

Problem: A card is drawn from a standard deck. Find the probability of getting a spade.

Solution:

Sample Space: Total cards = 52, n(S) = 52

Number of spades = 13, n(E) = 13

P(Spade) = 13/52 = 1/4 = 0.25

Example 3: Complementary Probability

Problem: In a class of 40 students, 25 like mathematics. What is the probability that a randomly selected student does not like mathematics?

Solution:

P(likes math) = 25/40 = 5/8

P(does not like math) = 1 - P(likes math) = 1 - 5/8 = 3/8

Important Points to Remember

Sample space must include ALL possible outcomes
Probability is always between 0 and 1 (inclusive)
Probability of mutually exclusive events depends on whether they can occur together
For equally likely outcomes, use: P(E) = n(E) / n(S)
The sum of probabilities of all simple events = 1
Complementary events are always mutually exclusive and exhaustive

📋 Board Exam Tips

Q: State the definition of random experiment clearly

Q: Distinguish between simple and compound events with examples

Q: Find probability using classical definition

Common Mistake: Students forget that probability must be ≤ 1. If you get P > 1, check your calculation!

🎯 JEE Advanced Tips

Key Concept: Master sample space construction for complex experiments (dice + cards, etc.)

Speed Tip: Use complementary probability when finding "at least one" problems

Common Pattern: Questions on probability often combine with permutations and combinations

Advanced Insight: Understanding sample spaces is crucial for conditional probability later

📐 Complete Formula Sheet

Probability of Event

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

Classical probability for equally likely outcomes

Complementary Probability

P(E') = 1 - P(E)

Probability of NOT E

Sure Event

P(S) = 1

Probability of sample space

Impossible Event

P(∅) = 0

Probability of empty set

Sum of All Probabilities

ΣP(Eᵢ) = 1

Sum of probabilities of all simple events equals 1

Range of Probability

0 ≤ P(E) ≤ 1

Probability is always non-negative and at most 1