Probability: Basic Concepts & Definitions

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What Is Probability?

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Key Vocabulary

1. Experiment

Example: Rolling a die, flipping a coin, drawing a card.

2. Sample Space (S)

Coin flip: S = {Heads, Tails} Rolling a die: S = {1, 2, 3, 4, 5, 6}

3. Event (E)

Event A = rolling an even number = {2, 4, 6}

4. Outcome

Rolling a 3 is one outcome.

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The Probability Formula (Classical)

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Types of Events

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Complement Rule

If P(rain) = 0.3, then P(no rain) = 0.7

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Three Approaches to Probability

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Quick Summary

• Probability lives between 0 and 1.
• The sample space lists every possible outcome.
• An event is any outcome (or group) you're measuring.
• Use P(E) = favorable / total when outcomes are equally likely.
• The complement of any event E has probability 1 − P(E).

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How to Find Probability

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Method 1: Classical Formula (Equally Likely Outcomes)

1. Count the total outcomes in the sample space
2. Count how many outcomes match your event
3. Divide

• Favorable outcomes: 4 Kings
• Total outcomes: 52 cards
• P(King) = 4/52 = 1/13 ≈ 0.077

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Method 2: Empirical Formula (Based on Data/Experiments)

• P(making a shot) = 35/50 = 0.70 = 70%

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Method 3: Complement (Easier Reverse Approach)

• P(no 6 on one die) = 5/6
• P(no 6 on either die) = 5/6 × 5/6 = 25/36
• P(at least one 6) = 1 − 25/36 = 11/36 ≈ 0.306

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Method 4: Addition Rule (Either/Or Events)

• P(King) = 4/52, P(Heart) = 13/52, P(King of Hearts) = 1/52
• P = 4/52 + 13/52 − 1/52 = 16/52 ≈ 0.308

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Method 5: Multiplication Rule (Both Events Together)

• P(1st Ace) = 4/52, P(2nd Ace | 1st was Ace) = 3/51
• P = 4/52 × 3/51 = 12/2652 ≈ 0.0045

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Quick Decision Guide

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When Favorable Outcomes = Total Outcomes

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Dice Example

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The 3 Special Cases to Know

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Simple Way to Think About It

If every outcome in the sample space satisfies your event, then no matter what happens, your event occurs — so probability = 1.

• Every possible result is between 1 and 6
• So it's guaranteed → P = 1

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Key Rule to Remember

• P = 0 → Impossible
• P = 1 → Certain
• P = 0.5 → Equal chance
• Anything between → Some level of likelihood

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Probability Practice Problems

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🎲 Basic Level

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🎯 Medium Level

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🔥 Challenge Level

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✅ Answers

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Answer 1

• Total balls = 3 + 2 + 5 = 10
• Favorable (red) = 3
• P = 3/10 = 0.3

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Answer 2

• Total outcomes = {H, T} = 2
• Favorable = {T} = 1
• P = 1/2 = 0.5

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Answer 3

• Total outcomes = {1,2,3,4,5,6} = 6
• Favorable (greater than 6) = none = 0
• P = 0/6 = 0 → Impossible ❌

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Answer 4

• Total outcomes = 6
• Favorable (even) = {2, 4, 6} = 3
• P = 3/6 = 1/2 = 0.5

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Answer 5 (Addition Rule)

• P(Queen) = 4/52
• P(Club) = 13/52
• P(Queen of Clubs) = 1/52 (overlap)
• P = 4/52 + 13/52 − 1/52 = 16/52 = 4/13 ≈ 0.308

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Answer 6 (Complement Rule)

• Total students = 12 + 8 = 20
• P(boy) = 12/20
• P(not boy) = 1 − 12/20 = 8/20 = 2/5 = 0.4

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Answer 7 (Complement Rule)

• P(no heads at all) = P(TT) = 1/2 × 1/2 = 1/4
• P(at least one Head) = 1 − 1/4 = 3/4 = 0.75

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Answer 8

• Total outcomes when rolling 2 dice = 6 × 6 = 36
• Combinations that sum to 7: (1,6)(2,5)(3,4)(4,3)(5,2)(6,1) = 6
• P = 6/36 = 1/6 ≈ 0.167

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Branches of Statistics

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🌐 Overview

STATISTICS
├── Descriptive Statistics
└── Inferential Statistics
        ├── Estimation
        └── Hypothesis Testing

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1. Descriptive Statistics

Describes and summarizes data in a meaningful way — no conclusions beyond the data itself.

What it does:

• Organizes raw data
• Presents data visually or numerically
• Answers: "What does the data look like?"

Tools Used:

Example:

A teacher finds the average score of 30 students on a test = 72/100 That's descriptive — it summarizes the class performance.

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2. Inferential Statistics

Uses a sample to make conclusions (inferences) about a larger population.

What it does:

• Draws conclusions beyond the data
• Makes predictions and generalizations
• Answers: "What can we conclude about the population?"

Two Main Sub-branches:

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2a. Estimation

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2b. Hypothesis Testing

"Is a new drug more effective than the old one?" You test data from a sample to accept or reject this claim.

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Other Notable Branches

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Key Difference: Descriptive vs Inferential

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Simple Analogy

Imagine you taste a spoon of soup to check if it's salty:

• Descriptive = "This spoon is salty" (describes what you observed)
• Inferential = "The whole pot must be salty" (conclusion about the bigger picture)

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Statistics vs Parameters

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Core Difference

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Definitions

📊 Parameter

A fixed numerical value that describes a characteristic of an entire population.

• It's the "truth" about the whole group
• Usually impossible to know exactly (too large to measure everyone)
• Represented by Greek letters

📋 Statistic

A numerical value calculated from a sample that is used to estimate a parameter.

• It's what we can actually measure
• Changes depending on which sample you pick
• Represented by Latin/Roman letters

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Symbol Table

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Real-World Examples

Example 1 — Student Exam Scores

A country has 5 million students who took a national exam.

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Example 2 — Election Poll

A city has 2 million voters.

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Example 3 — Factory Quality Control

A factory produces 100,000 bulbs per day.

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Visual Analogy

POPULATION  (everyone/everything)
     │
     │  ← Parameter describes this whole group
     │
  [Sample]  (a small selected group)
     │
     │  ← Statistic describes this sample
     │
     ▼
  Used to ESTIMATE the Parameter

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Key Relationship

We can't always measure the whole population, so we take a sample, calculate a statistic, and use it to estimate the parameter.

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Simple Memory Trick

P goes with P, S goes with S — that's all you need to remember!

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