Probability and Counting
Probability and counting quantify uncertainty and enumerate possibilities, forming the core of combinatorics and statistics. At MathMultiverse, we explore these concepts with clear formulas, engaging examples, and interactive visualizations, making them accessible for all learners.
From lottery odds to data analysis, these tools empower precise decision-making across diverse fields.
Formulas & Techniques
Probability
\[ P(E) = \frac{|E|}{|S|} \]
Permutations
\[ P(n, r) = \frac{n!}{(n - r)!} \]
Combinations
\[ C(n, r) = \frac{n!}{r! (n - r)!} \]
Addition Rule
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Conditional Probability
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
Bayes’ Theorem
\[ P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B | A) P(A) + P(B | A^c) P(A^c)} \]
Binomial Probability
\[ P(k) = \binom{n}{k} p^k (1 - p)^{n - k} \]
Examples
Basic Probability
Drawing an ace:
\[ P(\text{Ace}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769 \]
Permutations
Arrange 3 of 6 books:
\[ P(6, 3) = 6 \cdot 5 \cdot 4 = 120 \]
Combinations
Choose 2 cards from 5:
\[ C(5, 2) = 10 \]
Addition Rule
Heart or ace:
\[ P(\text{Heart} \cup \text{Ace}) = \frac{4}{13} \approx 0.3077 \]
Conditional Probability
Ace given heart:
\[ P(\text{Ace} | \text{Heart}) = \frac{1}{13} \approx 0.0769 \]
Binomial Probability
3 heads in 5 flips:
\[ P(3) = 10 \cdot 0.125 \cdot 0.25 = 0.3125 \]
Visualizations
Binomial Probability (5 Coin Flips)
Applications
- Gambling: Two aces in poker: \( \approx 0.0399 \).
- Statistics: Sample size combinations: \( C(100, 10) \).
- Risk Analysis: At least one system failure: \( 0.271 \).
- Cryptography: Key permutations: \( 26! \).
- Biology: Genotype combinations: \( C(4, 2) = 6 \).