Probability Calculator (Independent Events)
For independent events, where one result does not affect the next, calculate the probability that all happen, that none happen, and that at least one happens.
Formulas
When an event with probability p is tried n times, and each try is independent:
- All happen = p^n
- None happen = (1 − p)^n
- At least one happens = 1 − (1 − p)^n (the complement of "none happen")
When independent events have different probabilities p1, p2, …:
- All happen = p1 × p2 × …
- None happen = (1 − p1) × (1 − p2) × …
- At least one happens = 1 − (1 − p1) × (1 − p2) × …
The calculation uses exact fractions (BigInt), and values are rounded only for display (the percentage has 7 significant digits). A probability that does not terminate as a decimal, such as 1/6, is exact if you enter it as a fraction.
What "independent" means here
Two events are independent when knowing the result of one does not change the probability of the other. Rolling a die twice is independent. Drawing two cards from a deck without putting the first back is not, because the first draw changes what is left. Mutually exclusive events (for example, "heads" and "tails" in one flip) cannot both happen, so they are not independent, and "all happen" would be 0. This calculator does not handle either of those cases. Entering the same probabilities here does not make them independent.
Trials needed (reverse)
This finds the smallest number of trials for which the probability of "at least one" reaches your target. It is the smallest n for which (1 − p)^n is at most (1 − target), found by comparing exact fractions.
- The maximum is 1,000 trials. If the target is not reached within 1,000 trials, the result explains why and shows the probability at 1,000 trials.
- A probability of 0% can never happen, so any target above 0% cannot be reached.
- A target of 100% is never reached with a finite number of trials unless each trial has a probability of 100% (the probability only gets closer to 99.99…%). With 100% per trial, one trial is enough.
- A target of 0% is already met with no trials (0 trials).
Probabilities of 0%, 100% and 0 trials
- Probability 0%: "all happen" is 0 (for n ≥ 1), "none happen" is 1, "at least one" is 0.
- Probability 100%: "all happen" is 1, "none happen" is 0 (for n ≥ 1), "at least one" is 1.
- 0 trials: nothing is tried, so "all happen" and "none happen" are both 1 (100%), and "at least one" is 0. This is a special case.
Examples
- A 1/6 die rolled 3 times → all sixes = 1/216 (about 0.4630%); no six = 125/216 (about 57.87%); at least one six = 91/216 (about 42.13%)
- A 1% drop chance, 100 attempts → at least one drop about 63.40%; none about 36.60%
- Reverse: a 1% chance, with at least a 90% chance of at least one success → 230 trials (229 trials gives about 89.99%)
- Different probabilities 50%, 20%, 10% → all happen 1%; none happen 36%; at least one happens 64%
Entering a list of probabilities
Separate the probabilities with a line break, a space, a semicolon, or a comma followed by a space (for example, 50, 20, 10). A comma with no space after it is an error, because it could be a decimal comma (0,5) or a separator between two numbers, and the calculator does not guess. Write decimals with a period (0.5).
How this differs from permutations and combinations
The Permutation & Combination Calculator counts how many ways something can be chosen. This calculator finds the probability that events happen, when you already know how likely each trial is. To write a probability as "favorable outcomes ÷ all outcomes", you can count the outcomes with permutations and combinations and then use the Fraction Calculator.
The probability of exactly k successes (the binomial distribution) is not covered.
Frequently asked questions
How do I calculate "at least once"?
Subtract the probability of "none happen" from 1. For a 1% chance tried 100 times, the chance of never succeeding is 0.99^100 (about 36.60%), so the chance of at least one success is about 63.40%.
If something has a 1% chance, am I certain to get it in 100 tries?
No. There is about a 36.60% chance of getting it zero times. "1% × 100 tries" does not add up to 100%.
What if my events are not independent?
Then these formulas do not apply. For example, when you draw without replacement, each draw changes the probabilities of the next. This calculator does not handle that case.