Probability Basics

Probability quantifies uncertainty, enabling predictions in scenarios from games to AI. At MathMultiverse, we break down probability with clear definitions, engaging examples, interactive visualizations, and real-world applications, making it accessible for students and enthusiasts.

Why study probability? It’s the backbone of decision-making, risk analysis, and statistical modeling, shaping fields like finance, machine learning, and gaming.

Definition

Probability measures an event’s likelihood within a sample space of all possible outcomes. For an event \( A \):

\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]

Values range from 0 (impossible) to 1 (certain). This classical approach assumes equally likely outcomes, while other interpretations include long-term frequency or subjective belief.

Examples

Rolling a Die

For a six-sided die, the probability of rolling a 6 is:

\[ P(6) = \frac{1}{6} \approx 0.1667 \]

Probability of an even number (2, 4, 6):

\[ P(\text{even}) = \frac{3}{6} = 0.5 \]

Drawing a Card

In a 52-card deck, the probability of drawing an Ace is:

\[ P(\text{Ace}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769 \]

Probability of a heart:

\[ P(\text{heart}) = \frac{13}{52} = 0.25 \]

Flipping Coins

For two coins, the probability of two heads is:

\[ P(\text{HH}) = \frac{1}{4} = 0.25 \]

At least one head:

\[ P(\text{at least one head}) = \frac{3}{4} = 0.75 \]

Rolling Two Dice

Probability of a sum of 7:

\[ P(\text{sum = 7}) = \frac{6}{36} = \frac{1}{6} \approx 0.1667 \]

Picking Marbles

Bag with 5 red, 3 blue, 2 green marbles. Probability of red then blue (no replacement):

\[ P(\text{red then blue}) = \frac{5}{10} \times \frac{3}{9} = \frac{1}{6} \approx 0.1667 \]

Defective Items

100 items, 5 defective. Probability of both defective (no replacement):

\[ P(\text{both defective}) = \frac{5}{100} \times \frac{4}{99} \approx 0.002 \]

Visualizations

Die Outcomes

Two Coins

Sum of Two Dice

Rules

  • Addition Rule (Mutually Exclusive):
    \[ P(A \text{ or } B) = P(A) + P(B) \]

    Example: Rolling a 1 or 2: \( \frac{1}{3} \).

  • Complement Rule:
    \[ P(\text{not } A) = 1 - P(A) \]

    Example: Not rolling a 6: \( \frac{5}{6} \).

  • Multiplication Rule (Independent):
    \[ P(A \text{ and } B) = P(A) \times P(B) \]

    Example: Two heads: \( \frac{1}{4} \).

  • Multiplication Rule (Dependent):
    \[ P(A \text{ and } B) = P(A) \times P(B | A) \]

    Example: Red then blue marble: \( \frac{1}{6} \).

  • Conditional Probability:
    \[ P(B | A) = \frac{P(A \text{ and } B)}{P(A)} \]

    Example: Ace given heart: \( \frac{1}{13} \).

  • Total Probability:
    \[ P(A) = P(A | B_1) P(B_1) + P(A | B_2) P(B_2) \]

    Example: Defective item from two machines: \( 0.024 \).

Applications

  • Weather Forecasting: Probability of rain: \( 0.7 \), no rain: \( 0.3 \).
  • Insurance: Expected accident cost: $100.
  • Machine Learning: Spam detection, \( P(\text{spam}) = 0.9 \).
  • Medical Testing: Disease given positive test: \( \approx 0.323 \).
  • Gaming: Expected winnings: $0.
  • Quality Control: At least one defective: \( \approx 0.098 \).