Probability Calculator

Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). It's the foundation of statistics, gambling, insurance pricing, machine learning, and risk analysis. The simplest case — a single event with a finite list of equally likely outcomes — is just favorable outcomes divided by total outcomes.

Real problems usually involve combining events: do both happen, does either happen, what's the chance of A given that B already happened? Each of those questions has a specific rule. This calculator handles the four most common operations: a single event, AND (intersection), OR (union), and NOT (complement). For dependent events you'll need the conditional-probability form covered in the FAQ.

Single event. Rolling a 4 on a fair six-sided die:

• Favorable outcomes = 1 (just the face "4")
• Total outcomes = 6
• P(rolling a 4) = 1/6 ≈ 0.1667 (16.67%)

AND (independent). Flipping heads twice in a row:

• P(H) = 0.5 each flip
• P(H and H) = 0.5 × 0.5 = 0.25 (25%)

OR. Rolling a 1 or a 6 on a die (mutually exclusive):

• P(1) + P(6) − P(1 and 6) = 1/6 + 1/6 − 0 = 2/6 ≈ 0.333

Complement trick. Probability of at least one six in four rolls:

• P(no six in one roll) = 5/6
• P(no six in four rolls) = (5/6)⁴ ≈ 0.482
• P(at least one six) = 1 − 0.482 = 0.518 (51.8%)

• Games of chance — dice, cards, lotteries; fast sanity check on whether a bet is fair.
• Risk analysis — combine probabilities of failure events for a system or process.
• A/B testing — translate observed conversion rates into probabilities of seeing the result by chance.
• Insurance and pricing — premiums equal expected loss, which is loss × probability summed over scenarios.
• Decision-making under uncertainty — compute expected value before committing time or money.

• Multiplying probabilities of dependent events as if they were independent. Drawing two aces from a deck without replacement is 4/52 × 3/51, not (4/52)².
• Forgetting to subtract the overlap in P(A or B). Without it you'll get values above 1 and obvious nonsense.
• Confusing P(A|B) with P(B|A) — the gambler's fallacy and the prosecutor's fallacy both stem from this swap.
• Treating 'unlikely' as 'impossible'. A 1-in-100 event will happen on average 10 times in 1000 trials; rare events are routine over many tries.
• Adding probabilities for independent AND events. AND multiplies, OR adds (with overlap correction). Mixing them up is the most common error.