Probability

Master the fundamentals of probability theory and its applications

Table of Contents

Simplified Quantitative Formulas: Probability & Bayes' Theorem

  • Probability: Probability = (favorable outcomes) / (total outcomes), always between 0 and 1.
  • Independent Events: Two events are independent if the outcome of one does not affect the other (e.g., coin flips).
  • Dependent Events: If the occurrence of one event affects the probability of another (e.g., drawing cards without replacement).
  • Conditional Probability: P(B|A) = P(A ∩ B) / P(A). Out of cases where A occurs, how many also have B.
  • Bayes' Theorem: P(A|B) = [P(B|A) × P(A)] / P(B). Allows "reversing" conditional probability.
  • Sum Rule: For mutually exclusive events, P(A or B) = P(A) + P(B).
  • Product Rule: For independent events, P(A and B) = P(A) × P(B).
  • Complement: P(not A) = 1 – P(A).
  • Multiple Causes: For several causes, use extended Bayes' formula: P(Ak | B) = P(B|Ak)*P(Ak) / Σ_i [P(B|Ai)*P(Ai)].
  • Monty Hall/Dependent Example: Probability can change when new information is revealed (see Monty Hall problem).

What do these mean? (Super Simple Explanations & Examples)

  • Probability: If you toss a coin, the chance of heads is 1 out of 2, so probability = 1/2.
    Example: Rolling a die, probability of getting a 4 = 1/6.
  • Independent Events: Tossing two coins: the result of one does not affect the other.
    Example: Probability of getting heads on both = 1/2 × 1/2 = 1/4.
  • Dependent Events: Drawing two cards from a deck without replacement: the first draw affects the second.
    Example: Probability both are aces = 4/52 × 3/51.
  • Conditional Probability: Probability of B given A has happened.
    Example: In a class of 30, 20 boys, 10 girls, 5 boys wear glasses. Probability a student wears glasses given they are a boy = 5/20 = 0.25.
  • Bayes' Theorem: Used to update probability when new information is available.
    Example: If 1% have a disease, test is 90% accurate, Bayes' theorem tells you the chance someone has the disease if they test positive.
  • Sum Rule: Probability of A or B (if mutually exclusive) = P(A) + P(B).
    Example: Probability of rolling a 1 or 2 on a die = 1/6 + 1/6 = 1/3.
  • Product Rule: Probability of A and B (if independent) = P(A) × P(B).
    Example: Probability of getting two heads in two coin tosses = 1/2 × 1/2 = 1/4.
  • Complement: Probability of not getting a 6 on a die = 1 – 1/6 = 5/6.
  • Multiple Causes: If there are several possible causes for an event, use extended Bayes' formula.
    Example: Three factories supply bulbs with different defect rates; Bayes' theorem finds the chance a defective bulb came from a specific factory.
  • Monty Hall/Dependent Example: In the Monty Hall problem, switching doors increases your chance of winning from 1/3 to 2/3 after the host opens a door.

Introduction to Probability

Probability is the branch of mathematics that deals with the likelihood of events occurring. It helps us quantify uncertainty and make informed decisions based on data.

Key Concepts

  • Experiment: A process that produces outcomes
  • Sample Space: Set of all possible outcomes
  • Event: Subset of the sample space
  • Probability: Measure of likelihood (0 to 1)

Basic Probability Formula

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

where:

  • P(A) is the probability of event A
  • n(A) is the number of favorable outcomes
  • n(S) is the total number of possible outcomes

Basic Concepts

Sample Space

Examples:

  • Coin toss: S = {H, T}
  • Die roll: S = {1, 2, 3, 4, 5, 6}
  • Two coins: S = {HH, HT, TH, TT}

Events

Types of events:

  • Simple event: Single outcome
  • Compound event: Multiple outcomes
  • Impossible event: P(A) = 0
  • Certain event: P(A) = 1

Types of Events

Mutually Exclusive Events

Events that cannot occur simultaneously:

P(A ∩ B) = 0

P(A ∪ B) = P(A) + P(B)

Independent Events

Events where one does not affect the other:

P(A ∩ B) = P(A) × P(B)

Complementary Events

Events that are mutually exclusive and exhaustive:

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

Probability Rules

Addition Rule

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Multiplication Rule

P(A ∩ B) = P(A) × P(B|A)

Complement Rule

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

Conditional Probability

Definition

P(A|B) = P(A ∩ B)/P(B)

Properties

  • 0 ≤ P(A|B) ≤ 1
  • P(S|B) = 1
  • P(A'|B) = 1 - P(A|B)

Example

In a class of 30 students, 20 are boys and 10 are girls. 5 boys and 3 girls wear glasses. What is the probability that a student wears glasses given that they are a boy?

P(Glasses|Boy) = 5/20 = 0.25

Bayes' Theorem

Formula

P(A|B) = P(B|A) × P(A)/P(B)

Law of Total Probability

P(B) = Σ P(B|Aᵢ) × P(Aᵢ)

Example

A factory has two machines. Machine A produces 60% of items and Machine B produces 40%. 5% of items from Machine A are defective, while 3% from Machine B are defective. If an item is defective, what is the probability it came from Machine A?

P(A|D) = (0.05 × 0.6)/(0.05 × 0.6 + 0.03 × 0.4) = 0.714

Practice Questions

Basic Probability

Question 1: Simple Probability
A bag contains 5 red, 3 blue, and 2 green balls. What is the probability of drawing a red ball?
Solution:

Total balls = 10

Red balls = 5

Probability = 5/10 = 0.5

Question 2: Multiple Events
Two dice are rolled. What is the probability of getting a sum of 7?
Solution:

Favorable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

Total outcomes = 36

Probability = 6/36 = 1/6

Question 3: Complementary Events
The probability of rain tomorrow is 0.3. What is the probability that it will not rain?
Solution:

P(No rain) = 1 - P(Rain)

= 1 - 0.3 = 0.7

Question 4: Combined Probability
A box contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?
Solution:

P(First red) = 4/10

P(Second red) = 3/9

P(Both red) = (4/10) × (3/9) = 2/15

Conditional Probability

Question 1: Basic Conditional
In a class of 30 students, 20 are boys and 10 are girls. 5 boys and 3 girls wear glasses. What is the probability that a student wears glasses given that they are a boy?
Solution:

P(Glasses|Boy) = 5/20 = 0.25

Question 2: Multiple Conditions
A bag contains 3 red and 4 blue balls. Two balls are drawn without replacement. What is the probability that the second ball is red given that the first was blue?
Solution:

After first draw: 3 red, 3 blue balls

P(Second red|First blue) = 3/6 = 0.5

Question 3: Independent Events
A coin is tossed twice. What is the probability of getting heads both times?
Solution:

P(First head) = 1/2

P(Second head) = 1/2

P(Both heads) = (1/2) × (1/2) = 1/4

Question 4: Dependent Events
A deck of 52 cards is shuffled. What is the probability of drawing two aces without replacement?
Solution:

P(First ace) = 4/52

P(Second ace) = 3/51

P(Two aces) = (4/52) × (3/51) = 1/221

Bayes' Theorem

Question 1: Basic Bayes
A factory has two machines. Machine A produces 60% of items and Machine B produces 40%. 5% of items from Machine A are defective, while 3% from Machine B are defective. If an item is defective, what is the probability it came from Machine A?
Solution:

P(A|D) = (0.05 × 0.6)/(0.05 × 0.6 + 0.03 × 0.4)

= 0.714

Question 2: Multiple Sources
A company has three suppliers. Supplier A provides 50% of parts, B provides 30%, and C provides 20%. The defect rates are 2%, 3%, and 4% respectively. If a part is defective, what is the probability it came from Supplier B?
Solution:

P(B|D) = (0.03 × 0.3)/(0.02 × 0.5 + 0.03 × 0.3 + 0.04 × 0.2)

= 0.333

Question 3: Medical Test
A medical test is 95% accurate. 1% of the population has the disease. If a person tests positive, what is the probability they actually have the disease?
Solution:

P(D|T+) = (0.95 × 0.01)/(0.95 × 0.01 + 0.05 × 0.99)

= 0.161

Question 4: Weather Forecast
A weather forecast is 80% accurate. The probability of rain is 30%. If the forecast predicts rain, what is the probability it will actually rain?
Solution:

P(R|F) = (0.8 × 0.3)/(0.8 × 0.3 + 0.2 × 0.7)

= 0.632

Practice Questions

Test your understanding of Probability with 20 fully solved, step-by-step questions designed for beginners.

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