Probability calculator

Probability measures the chance of an event, from 0 (impossible) to 1 (certain) — or 0% to 100%. In the classic form:

P = favorable cases ÷ total possible cases

Step-by-step example

Rolling a 5 on a 6-sided die: 1 favorable case in 6 → 1/6 ≈ 16.67%.

Combined events

• "AND" (independent): multiply. Two 6s in a row: 1/36.
• "OR" (exclusive): add. A 5 or 6: 1/3.
• Complement: P(not happening) = 1 − P(happening).

The complement trick

Questions of the form "what is the chance of at least one…" are almost always easier backwards. Computing the chance of none and subtracting from 1 avoids adding up dozens of overlapping cases.

What is the chance of rolling at least one 6 in four throws of a die? Directly, that is four cases with overlaps. By complement: the chance of not rolling a 6 once is 5/6; four times in a row, (5/6)⁴ = 0.4823. So at least one 6 has a 51.8% chance.

The gambler's fallacy

Independent events have no memory. If a coin has come up heads five times, the chance of tails next is still 50% — the coin does not know what happened before. The intuition that tails is "due" is the gambler's fallacy, and it is the same one that underwrites betting systems which only work with infinite money.