Probability Calculator

What are the chances? Probability quantifies uncertainty — from coin flips to medical tests to risk analysis. This calculator handles single events, multiple independent events, conditional probability, and probability distributions.

How to Use This Calculator

1. Select the probability type: single event, multiple events (AND/OR), conditional, or binomial.
2. Enter the number of favorable outcomes and total possible outcomes.
3. For multiple events: enter probabilities for each and specify AND (both occur) or OR (at least one occurs).
4. Click Calculate to see probability as a fraction, decimal, and percentage.

Core Probability Formulas

Basic probability: P(A) = favorable outcomes / total outcomes

AND (both events): P(A ∩ B) = P(A) × P(B) if independent

OR (at least one): P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Complement: P(not A) = 1 − P(A)

Conditional: P(A|B) = P(A ∩ B) / P(B)

Example Calculations

Rolling a 6 on a die: 1/6 ≈ 16.67%

Rolling 6 twice in a row: (1/6) × (1/6) = 1/36 ≈ 2.78%

Rolling at least one 6 in two rolls: 1 − (5/6)² = 1 − 25/36 = 11/36 ≈ 30.56%

Drawing a heart from a deck: 13/52 = 1/4 = 25%

Probability vs. Odds

• Probability: favorable / total. P = 1/6 for rolling a 6.
• Odds for: favorable : unfavorable = 1:5 (one way to win, five ways to lose).
• Odds against: unfavorable : favorable = 5:1.

Common Mistakes to Avoid

• Assuming independent events when they're not — Drawing cards without replacement changes the probability with each draw. Always check if events affect each other.
• Adding instead of using the OR formula — P(A or B) ≠ P(A) + P(B) when events can overlap. Use P(A) + P(B) − P(A and B).
• Confusing "at least one" with "exactly one" — These are different. "At least one" uses the complement rule; "exactly one" requires more careful calculation.

Frequently Asked Questions

What is the law of large numbers?

As a random experiment is repeated many times, the observed frequency of an outcome approaches its theoretical probability. A fair coin won't always be 50/50 in 10 flips, but over 10,000 flips it'll be very close.

What is expected value?

The long-run average outcome of a random variable: EV = Σ(outcome × probability). A lottery ticket with 1% chance of winning $100: EV = $1. If the ticket costs $2, it's a losing bet on average.

What is Bayes' theorem?

P(A|B) = P(B|A) × P(A) / P(B). It updates probability estimates as new evidence arrives. Used in medical testing, spam filters, and machine learning. A positive medical test doesn't mean you're definitely sick — Bayes shows you how to calculate the true probability given test accuracy and disease prevalence.

Conclusion

Probability thinking helps you make better decisions under uncertainty — whether you're evaluating risk, playing games, or interpreting medical test results. Use this calculator to quantify any probability and understand the mathematics of chance.

Related: Statistics Calculator | Random Number Generator | Sample Size Calculator | Z-Score Calculator