Probability Calculator

Probability is one of the few areas of mathematics where human intuition is reliably, spectacularly wrong. We overestimate rare dramatic events and underestimate how quickly small chances compound.

Entering your probabilities

1. Enter the probability of event A as a decimal or a percentage.
2. Enter the probability of event B if you are combining two events.
3. Read the chance of both occurring, either occurring, and neither occurring.
4. Check the complement — the chance the event does not happen.
5. Remember these formulas assume the events are independent.

The core rules

• Single probability — favourable outcomes divided by total possible outcomes. One specific card from a deck is 1 ÷ 52.
• Both events (AND) — multiply the probabilities. Two coin flips both landing heads is 0.5 × 0.5 = 0.25.
• Either event (OR) — add them, then subtract the overlap so it is not counted twice.
• Complement — one minus the probability. If something has a 30% chance, there is a 70% chance it does not happen.
• Independence — event A not affecting event B. Coin flips are independent; drawing cards without replacement is not.

How small risks stack up

A 1% chance of failure sounds negligible. Run the same process a hundred times and the chance that everything succeeds is 0.99 to the power of 100 — about 36.6%. In other words, failure somewhere along the line is the likely outcome, not the unlikely one.

This is the arithmetic behind reliability engineering, and behind why long chains of dependencies break so often. Each individual link looks safe; the chain does not.

Common traps

• Assuming independence when events actually influence each other.
• The gambler’s fallacy: a coin has no memory, and five heads do not make tails more likely.
• Confusing the probability of A given B with the probability of B given A.
• Ignoring base rates, which is what makes rare-disease test results so counter-intuitive.
• Adding probabilities for overlapping events without subtracting the overlap.

Probability questions