Probability Calculator

When to use it

Probability is the arithmetic of uncertainty, and the practical version has two parts: how likely one attempt is, and how likely at least one success is across several attempts. Dice and card games are the obvious cases; sales calls, quality sampling, email replies and repeated tests follow the same logic.

The multi-attempt figure assumes each try is independent and identical, which is true for dice and false for drawing cards without replacement. Getting that assumption wrong is the most common way these numbers mislead.

How the maths works

Single attempt: p = favourable outcomes ÷ total outcomes, printed as a percentage with four decimals. Odds against are total ÷ favourable, expressed as "about 18 to 1".

At least one success in k attempts: 1 − (1 − p)^k. The bracket is the chance of failing every time, so subtracting it from one leaves the chance of at least one success. Note what is missing: the formula never predicts exactly one success, only one or more.

A worked example

Rolling a total of 7 with two dice: 6 favourable outcomes out of 36.

1. One roll: 6 ÷ 36 = 0.055556, so 5.5556%.
2. Odds against, as the page reports them: 36 ÷ 6 = 6, so about 6 to 1.
3. Ten independent rolls: 1 − 0.944444^10 = 0.43537, so 43.537%.
4. Six rolls of a single six: 1 − (5/6)^6 = 66.5102%.

Repeated attempts

Check the first row: 1 − (5/6)^3 = 1 − 0.5787 = 0.4213. Doubling the attempts never doubles the chance; it closes the gap to certainty more and more slowly.

Common mistakes

• Assuming independence. Cards drawn without replacement change the odds after every draw, so the multi-attempt formula does not apply.
• Reading "at least one" as "exactly one". Ten rolls of two dice give 43.5% for one or more sevens, not for a single seven.
• Adding probabilities. Six attempts at 16.67% is 66.5%, not 100%; adding them would break the ceiling of certainty.
• Confusing odds with probability. Odds of 5 to 1 against mean one win in six, not 5 in 6.

Keep exploring

• Odds to Probability Calculator — the same information written as odds.
• Dice Roller — roll the dice this page is often used with.
• Statistics Calculator — describe the results once you have data.