Module 3 Study Guide

Probability Basics

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1. Introduction to Probability

Probability: A numerical measure (0 to 1) of the likelihood that an event will occur.

Key Terminology

Experiment: Any process that generates well-defined outcomes
Sample Space (S): The set of all possible outcomes
Event (A): A collection of one or more outcomes from the sample space

Classical Probability Formula

P(A) = n(A) / n(S)

Where: n(A) = outcomes in event A, n(S) = total outcomes in S

Fundamental Probability Rules

Rule Formula Meaning
Range 0 ≤ P(A) ≤ 1 All probabilities between 0 and 1
Certainty P(S) = 1 Something in sample space must occur
Impossibility P(∅) = 0 Impossible events have probability 0
Complement P(not A) = 1 − P(A) Probability A doesn't occur

2. Counting Methods

Fundamental Counting Principle (FCP)

If event 1 can occur in m ways and event 2 in n ways:

Total ways = m × n

Extended: n₁ × n₂ × n₃ × ...

Factorials

n! = n × (n−1) × (n−2) × ... × 3 × 2 × 1

Special case: 0! = 1

Permutations (Order Matters)

Arranging all n objects:

n!

Arranging r objects from n:

P(n,r) = n! / (n−r)!

Shortcut: n × (n−1) × (n−2) × ... × (n−r+1)

Combinations (Order Doesn't Matter)

Choosing r objects from n:

C(n,r) = n! / (r! × (n−r)!)

Relationship: C(n,r) = P(n,r) / r!

When to Use Which Method

Situation Does Order Matter? Use
Seating arrangement YES Permutation
Selecting committee members NO Combination
Race winners (1st, 2nd, 3rd) YES Permutation
Lottery numbers NO Combination
Password letters YES Permutation
Key Question: "If I swap two items, do I get a different outcome?"
YES → Permutation | NO → Combination

3. Conditional Probability & Independence

Conditional Probability

P(A|B): The probability of A given that B has occurred

P(A|B) = P(A and B) / P(B)

Where: P(B) > 0

Multiplication Rule

General Form:

P(A and B) = P(A) × P(B|A)

OR

P(A and B) = P(B) × P(A|B)

Independent Events

Independent: Events A and B are independent if the occurrence of one does NOT affect the probability of the other.
Test for Independence:
A and B are independent if P(A|B) = P(A)
Equivalently: P(B|A) = P(B)
Or: P(A and B) = P(A) × P(B)

For Independent Events:

P(A and B) = P(A) × P(B)

Independent Dependent
Flipping two coins Drawing cards WITHOUT replacement
Rolling two dice Weather on consecutive days
Drawing WITH replacement Exam scores and study time

4. Probability Distributions

Random Variables

Random Variable (X): A variable whose value is determined by the outcome of a random experiment
Discrete: Can only take specific, countable values (0, 1, 2, ...)
Continuous: Can take any value in a range (infinite possibilities)

Probability Distribution

Probability Distribution: Lists all possible values of a random variable with their probabilities

Requirements for Valid Distribution

  1. All probabilities between 0 and 1: 0 ≤ P(X = x) ≤ 1
  2. Probabilities sum to 1: ΣP(X = x) = 1

Expected Value (Mean)

E(X) = μ = Σ[x · P(X = x)]

In words: Multiply each value by its probability, then sum

Expected Value: The long-run average value if the experiment is repeated many times. It doesn't have to be a possible value of X!

Variance and Standard Deviation

Variance:

Var(X) = σ² = Σ[(x − μ)² · P(X = x)]

Standard Deviation:

SD(X) = σ = √Var(X)

Quick Reference: All Formulas

Basic Probability

P(A) = n(A) / n(S)

P(not A) = 1 − P(A)

Counting Methods

FCP: n₁ × n₂ × n₃ × ...

Factorial: n!

Permutation: P(n,r) = n! / (n−r)!

Combination: C(n,r) = n! / (r!(n−r)!)

Conditional Probability

P(A|B) = P(A and B) / P(B)

P(A and B) = P(A) × P(B|A)

Independent: P(A and B) = P(A) × P(B)

Probability Distributions

ΣP(X) = 1

E(X) = Σ[x · P(x)]

Var(X) = Σ[(x − μ)² · P(x)]

SD(X) = √Var(X)

Key Reminders

1. All probabilities: 0 ≤ P(A) ≤ 1
2. P(not A) = 1 − P(A) (complement rule)
3. Order matters → Permutation; Order doesn't matter → Combination
4. Independent: P(A and B) = P(A) × P(B)
5. Expected value = long-run average
6. WITHOUT replacement → dependent; WITH replacement → independent

Module 3: Probability Basics

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