🎲 Counting & Probability — Formulas

Essential formulas and shortcuts for working with counting and probability in MATHCOUNTS.

Basic Counting Formulas

Fundamental Counting Principle

Formula: If there are m ways to do one thing and n ways to do another, then there are m × n ways to do both Example: 3 shirts × 4 pants = 12 outfits

Addition Principle

Formula: If there are m ways to do one thing and n ways to do another, and these are mutually exclusive, then there are m + n ways to do either Example: 3 red shirts + 4 blue shirts = 7 shirts

Multiplication Principle

Formula: If there are m ways to do one thing, and for each of these ways there are n ways to do another, then there are m × n ways to do both Example: 3 ways to choose shirt × 4 ways to choose pants = 12 ways to choose both

Permutation Formulas

Basic Permutations

Formula: P(n, r) = n! / (n - r)! Example: P(5, 3) = 5! / (5 - 3)! = 5! / 2! = 120 / 2 = 60

Permutations with Repetition

Formula: n^r Example: 3-digit numbers using digits 1-5 with repetition: 5³ = 125

Permutations with Restrictions

Formula: Use multiplication principle and adjust for restrictions Example: 5 people sit in row with 2 specific people together: 4! × 2! = 48

Circular Permutations

Formula: (n - 1)! Example: 5 people sit in circle: (5 - 1)! = 4! = 24

Combination Formulas

Basic Combinations

Formula: C(n, r) = n! / (r!(n - r)!) Example: C(5, 3) = 5! / (3!2!) = 120 / (6 × 2) = 10

Combinations with Repetition

Formula: C(n + r - 1, r) Example: 3 identical balls in 5 boxes: C(5 + 3 - 1, 3) = C(7, 3) = 35

Combinations with Restrictions

Formula: Use addition principle and adjust for restrictions Example: 3 people from 5 with 2 specific people not both: C(5, 3) - C(3, 1) = 7

Probability Formulas

Basic Probability

Formula: P(event) = Number of favorable outcomes / Total number of outcomes Example: P(rolling 3) = 1/6

Complementary Probability

Formula: P(not A) = 1 - P(A) Example: P(not rolling 3) = 1 - 1/6 = 5/6

Independent Events

Formula: P(A and B) = P(A) × P(B) Example: P(rolling 3 and 4) = (1/6) × (1/6) = 1/36

Dependent Events

Formula: P(A and B) = P(A) × P(B|A) Example: P(two red cards) = (26/52) × (25/51) = 25/102

Mutually Exclusive Events

Formula: P(A or B) = P(A) + P(B) Example: P(rolling 3 or 4) = 1/6 + 1/6 = 1/3

Non-Mutually Exclusive Events

Formula: P(A or B) = P(A) + P(B) - P(A and B) Example: P(rolling even or 3) = 3/6 + 1/6 - 0 = 2/3

Conditional Probability Formulas

Basic Conditional Probability

Formula: P(A|B) = P(A and B) / P(B) Example: P(red|heart) = (13/52) / (13/52) = 1

Bayes’ Theorem

Formula: P(A|B) = P(B|A) × P(A) / P(B) Example: P(disease|positive) = P(positive|disease) × P(disease) / P(positive)

Law of Total Probability

Formula: P(B) = P(B|A₁) × P(A₁) + P(B|A₂) × P(A₂) + … + P(B|Aₙ) × P(Aₙ) Example: P(positive) = P(positive|disease) × P(disease) + P(positive|no disease) × P(no disease)

Expected Value Formulas

Basic Expected Value

Formula: E(X) = Σ(x × P(x)) Example: E(rolling die) = 1 × (1/6) + 2 × (1/6) + … + 6 × (1/6) = 3.5

Linearity of Expected Value

Formula: E(aX + bY) = aE(X) + bE(Y) Example: E(2X + 3Y) = 2E(X) + 3E(Y)

Expected Value of Independent Variables

Formula: If X and Y are independent, then E(XY) = E(X)E(Y) Example: If E(X) = 3 and E(Y) = 4, then E(XY) = 3 × 4 = 12

Probability Distribution Formulas

Binomial Distribution

Formula: P(X = k) = C(n, k) × p^k × (1 - p)^(n - k) Example: P(3 heads in 5 flips) = C(5, 3) × (1/2)³ × (1/2)² = 5/16

Geometric Distribution

Formula: P(X = k) = (1 - p)^(k - 1) × p Example: P(first head on 4th flip) = (1/2)³ × (1/2) = 1/16

Hypergeometric Distribution

Formula: P(X = k) = C(K, k) × C(N - K, n - k) / C(N, n) Example: P(2 red cards in 5 draws) = C(26, 2) × C(26, 3) / C(52, 5) ≈ 0.325

Common Counting Patterns

Arrangements

Linear arrangements: n! ways to arrange n distinct objects Circular arrangements: (n - 1)! ways to arrange n distinct objects in a circle Arrangements with restrictions: Use multiplication principle

Examples:

  • 5 people in row: 5! = 120
  • 5 people in circle: 4! = 24
  • 5 people in row with 2 together: 4! × 2! = 48

Selections

Combinations: C(n, r) ways to choose r objects from n Combinations with repetition: C(n + r - 1, r) ways to choose r objects from n with repetition Selections with restrictions: Use addition principle

Examples:

  • 3 people from 5: C(5, 3) = 10
  • 3 identical balls in 5 boxes: C(7, 3) = 35
  • 3 people from 5 with restrictions: Use addition principle

Distributions

Distinct objects to distinct boxes: n^r ways to distribute r distinct objects to n distinct boxes Identical objects to distinct boxes: C(n + r - 1, r) ways to distribute r identical objects to n distinct boxes Distinct objects to identical boxes: Use Stirling numbers (advanced)

Examples:

  • 3 distinct balls in 5 boxes: 5³ = 125
  • 3 identical balls in 5 boxes: C(7, 3) = 35
  • 3 distinct balls in 5 identical boxes: Use Stirling numbers

Mental Math Shortcuts

Common Combinations

C(n, 0) = 1: Choosing 0 objects C(n, 1) = n: Choosing 1 object C(n, n) = 1: Choosing all objects C(n, r) = C(n, n - r): Symmetry property

Examples:

  • C(5, 0) = 1
  • C(5, 1) = 5
  • C(5, 5) = 1
  • C(5, 2) = C(5, 3) = 10

Common Permutations

P(n, 0) = 1: Arranging 0 objects P(n, 1) = n: Arranging 1 object P(n, n) = n!: Arranging all objects P(n, r) = n × (n-1) × … × (n-r+1): Direct calculation

Examples:

  • P(5, 0) = 1
  • P(5, 1) = 5
  • P(5, 5) = 5! = 120
  • P(5, 3) = 5 × 4 × 3 = 60

Common Probabilities

P(impossible event) = 0: Event cannot happen P(certain event) = 1: Event must happen P(complement) = 1 - P(event): Probability of not happening P(independent events) = P(A) × P(B): Probability of both happening

Examples:

  • P(rolling 7 on die) = 0
  • P(rolling 1-6 on die) = 1
  • P(not rolling 3) = 1 - 1/6 = 5/6
  • P(rolling 3 and 4) = (1/6) × (1/6) = 1/36

Common Applications

Games and Gambling

Dice games: Use probability to calculate odds Card games: Use combinations to count hands Lottery: Use combinations to calculate winning probabilities

Examples:

  • Probability of rolling 7: 6/36 = 1/6
  • Number of poker hands: C(52, 5) = 2,598,960
  • Probability of winning lottery: 1/C(49, 6) ≈ 1/13,983,816

Statistics

Sampling: Use combinations to count samples Hypothesis testing: Use probability to make decisions Confidence intervals: Use probability to estimate parameters

Examples:

  • Number of samples of size 10 from 100: C(100, 10)
  • P-value for hypothesis test: Use probability distribution
  • 95% confidence interval: Use probability theory

Real-world Problems

Quality control: Use probability to test products Risk assessment: Use probability to evaluate risks Decision making: Use expected value to make choices

Examples:

  • Probability of defective product: Use binomial distribution
  • Expected value of investment: Use expected value formula
  • Risk of failure: Use probability calculations

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