Probability Calculator

Probability is a branch of mathematics that quantifies the likelihood of an event occurring, expressed as a number between 0 and 1, or as a percentage between 0% and 100%. Core Definition: Probability measures how likely an event is to happen. Basic Formula: P(A) = Number of favorable outcomes / Total number of possible outcomes Where: • P(A) = probability of event A • 0 ≤ P(A) ≤ 1 • P(A) = 0: Event is impossible • P(A) = 1: Event is certain • P(A) = 0.5: Event has equal chance of occurring or not Expressing Probability: • Fraction: 3/4 • Decimal: 0.75 • Percentage: 75% • Ratio/Odds: 3:1 (3 favorable to 1 unfavorable) Fundamental Rules: 1. Complement Rule: P(not A) = 1 - P(A) Example: If P(rain) = 0.3, then P(no rain) = 0.7 2. Addition Rule (OR): For mutually exclusive events: P(A or B) = P(A) + P(B) For non-mutually exclusive events: P(A or B) = P(A) + P(B) - P(A and B) 3. Multiplication Rule (AND): For independent events: P(A and B) = P(A) × P(B) For dependent events: P(A and B) = P(A) × P(B|A) 4. Conditional Probability: P(A|B) = P(A and B) / P(B) Read as: "Probability of A given B" Types of Events: • Independent: One event doesn't affect the other Example: Flipping coin twice • Dependent: One event affects the probability of another Example: Drawing cards without replacement • Mutually Exclusive: Cannot occur simultaneously Example: Rolling a 2 or 5 on one die Combinatorics: 1. Permutations (Order Matters): P(n,r) = n! / (n-r)! Arranging r items from n items 2. Combinations (Order Doesn't Matter): C(n,r) = n! / [r!(n-r)!] Selecting r items from n items Where n! = n × (n-1) × (n-2) × ... × 2 × 1 Common Probability Distributions: 1. Binomial Distribution: For n trials, probability of exactly k successes: P(X = k) = C(n,k) × p^k × (1-p)^(n-k) 2. Normal Distribution: Bell curve, characterized by mean and standard deviation Key Properties: • All probabilities sum to 1: ΣP(all outcomes) = 1 • Law of Large Numbers: As trials increase, experimental probability approaches theoretical • Sample space: Set of all possible outcomes Common Examples: • Coin flip: P(heads) = 1/2 = 0.5 = 50% • Die roll: P(any number) = 1/6 ≈ 0.167 = 16.7% • Deck of cards: P(ace) = 4/52 = 1/13 ≈ 0.077 = 7.7% Applications: Games of chance, weather forecasting, risk assessment, insurance, quality control, genetics, medical diagnostics, stock market analysis, sports predictions, and decision-making under uncertainty.