🎲 Probability Calculator
Calculate probabilities for single events, compound events, and conditional probability
Single Event Probability
P(E) = Favorable / Total
Complementary Probability
P(E') = 1 - P(E)
Compound Probability (AND)
P(A and B) = P(A) × P(B) for independent events
Compound Probability (OR)
P(A or B) = P(A) + P(B) - P(A and B)
Conditional Probability
P(A|B) = P(A and B) / P(B)
Dice Roll Probability
Calculate probability of rolling specific numbers
Essential Probability Formulas
Basic Probability
Complement Rule
Compound Events
Conditional Probability
How to Use This Calculator
- Single Event: Enter favorable outcomes and total possible outcomes to calculate basic probability
- Complement: Enter a probability to find the probability of the event NOT occurring
- AND (both events): Enter probabilities for two independent events occurring together
- OR (either event): Enter probabilities and overlap to find probability of at least one event
- Conditional: Enter P(A and B) and P(B) to find probability of A given B occurred
- Dice: Enter number of dice and target sum to calculate rolling probability
- All results show both percentage and decimal formats for convenience
- Results update automatically as you change any input value
Understanding Probability
What is Probability?
Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1 (or 0% to 100%). A probability of 0 means the event is impossible, while 1 means it's certain. Most events fall somewhere in between. The basic formula is P(E) = (number of favorable outcomes) / (total possible outcomes). Probability is everywhere: weather forecasts (70% chance of rain), games of chance (lottery odds), medical diagnoses (test accuracy), quality control (defect rates), and insurance (risk assessment). Understanding probability helps you make informed decisions under uncertainty.
Independent vs. Dependent Events
Independent events: The outcome of one event doesn't affect the probability of the other. Examples include flipping two coins, rolling dice multiple times (with replacement), or different people taking tests. For independent events, multiply probabilities: P(A and B) = P(A) × P(B). Dependent events: The first event affects the probability of the second. Examples include drawing cards without replacement, selecting items without putting them back, or conditional scenarios. For dependent events, use P(A and B) = P(A) × P(B|A), where P(B|A) is the conditional probability of B given A occurred. Recognizing this distinction is crucial for accurate probability calculations.
Conditional Probability & Bayes' Theorem
Conditional probability P(A|B) measures the likelihood of event A occurring given that B has already occurred. The formula P(A|B) = P(A and B) / P(B) effectively restricts our sample space to only outcomes where B happened. This is essential in medical testing (probability of disease given positive test), machine learning (classification based on features), and decision-making (updated probabilities with new information). Bayes' Theorem extends this concept: P(A|B) = P(B|A) × P(A) / P(B). This allows us to "reverse" conditional probabilities and update our beliefs based on evidence, forming the foundation of Bayesian statistics.
Common Probability Pitfalls
1) Gambler's Fallacy: Believing past independent events affect future ones ("I've flipped 5 heads, so tails is 'due'"). Each flip is still 50-50. 2) Ignoring base rates: Focusing on conditional probabilities without considering how common the condition is overall. 3) Confusion of the inverse: Thinking P(A|B) equals P(B|A). These are different! 4) Assuming independence: Multiplying probabilities for dependent events. 5) Sample size neglect: Not recognizing that larger samples give more reliable probability estimates. Understanding these helps avoid common mistakes in probability reasoning.
Tips for Probability Success
1) Draw diagrams: Tree diagrams, Venn diagrams, and tables help visualize probability problems. 2) Check your answer: Probabilities must be between 0 and 1. If not, you made an error. 3) Use complements: Sometimes it's easier to calculate P(not E) and subtract from 1. 4) List outcomes: For small sample spaces, enumerate all possibilities systematically. 5) Identify independence: Determine if events are independent or dependent before choosing a formula. 6) Practice with real scenarios: Apply probability to everyday situations to build intuition. 7) Learn fundamental counting: Permutations and combinations help count outcomes in complex scenarios.
Frequently Asked Questions
What is probability in statistics?
Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1 (or 0% to 100%). A probability of 0 means impossible, 1 means certain, and 0.5 means equally likely to happen or not. It's calculated as P(E) = (number of favorable outcomes) / (total possible outcomes). Probability is fundamental to statistics, helping quantify uncertainty in experiments, surveys, and real-world scenarios.
How do you calculate compound probability?
For independent events: AND (both occur) uses multiplication: P(A and B) = P(A) × P(B). Example: probability of two heads in a row = 0.5 × 0.5 = 0.25. OR (at least one occurs) uses addition: P(A or B) = P(A) + P(B) - P(A and B). The subtraction prevents double-counting when both events can occur simultaneously. For dependent events, adjust the second probability based on the first outcome.
What is conditional probability?
Conditional probability P(A|B) measures the likelihood of event A occurring given that event B has already occurred. Formula: P(A|B) = P(A and B) / P(B). This effectively restricts the sample space to only outcomes where B occurred. Example: Given a card is red (B), what's the probability it's a heart (A)? P(heart|red) = P(heart and red) / P(red) = (13/52) / (26/52) = 0.5.
What is complementary probability?
The complement of an event E (written E' or E̅) is the event NOT occurring. Since an event either happens or doesn't, P(E) + P(E') = 1, so P(E') = 1 - P(E). Example: if probability of rain is 0.3, probability of no rain is 1 - 0.3 = 0.7 or 70%. Complements are useful for calculating 'at least one' scenarios: P(at least one) = 1 - P(none).
What is the difference between independent and dependent events?
Independent events: one event's occurrence doesn't affect the other's probability. Example: flipping two coins—the first flip doesn't influence the second. Use simple multiplication: P(A and B) = P(A) × P(B). Dependent events: one event affects the other's probability. Example: drawing cards without replacement—first draw changes probabilities for the second. Must use conditional probability: P(A and B) = P(A) × P(B|A).
Is this probability calculator free to use?
Yes! This probability calculator is completely free with no hidden costs, subscriptions, or limitations. Use it for homework, statistics courses, or learning probability concepts.
Is my data private?
Absolutely. All processing happens locally in your browser. Your data never leaves your device and is not stored on any server.