🎲 Probability Calculator
Probability of A and B, A or B, neither, at least once in n tries, or exactly k successes in n trials. Accepts decimals, percents and fractions.
Enter probabilities as a decimal (0.25), a percent (25%) or a fraction (1/4).
Assuming A and B are independent (one does not affect the other):
P(A and B) = P(A) × P(B) = 0.5 × 0.16666667 · P(A or B) = P(A) + P(B) − P(A and B).
What Probability Calculator Does
This probability calculator answers the most common probability questions in one place. For two independent events it gives the chance of both, either, exactly one, and neither. For repeated tries it gives the chance of something happening at least once — and how many tries you need to be 50%, 90%, 95% or 99% sure. For repeated trials with a fixed chance, the binomial mode gives the probability of exactly, at most, at least, fewer than or more than k successes.
Enter probabilities however you think of them: 0.25, 25% or 1/4. Every answer is shown as a percentage, a decimal and as “1 in N” odds, with the formula underneath so you can check the reasoning.
How to Use Probability Calculator
- Choose two events, at least once, or binomial
- Enter probabilities as decimals, percents or fractions
- For binomial, enter the number of trials and successes
- Read the results as percentages, decimals and odds
- Check the formula line to see how each answer is found
Formula Used by Probability Calculator
Two independent events
P(A and B) = P(A) × P(B) · P(A or B) = P(A) + P(B) − P(A) × P(B)
Worked example
A coin lands heads (1/2) and a die shows six (1/6).
- Both: 1/2 × 1/6 = 1/12
- Either: 1/2 + 1/6 − 1/12 = 7/12
Result: 8.33% for both, 58.3% for at least one.
At least once in n tries
P = 1 − (1 − p)ⁿ
Worked example
At least one six in 4 rolls.
- (5/6)⁴ = 0.482
- 1 − 0.482
Result: 51.8% — just better than even.
Binomial
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
Worked example
Exactly 3 heads in 10 fair coin flips.
- C(10, 3) = 120
- 120 × 0.5³ × 0.5⁷ = 120 ÷ 1,024
Result: 11.7%.
Chance of at Least One Success
| Chance each time | 5 tries | 10 tries | 20 tries |
|---|---|---|---|
| 1% (1 in 100) | 4.9% | 9.6% | 18.2% |
| 5% (1 in 20) | 22.6% | 40.1% | 64.2% |
| 10% (1 in 10) | 41.0% | 65.1% | 87.8% |
| 1/6 (a die) | 59.8% | 83.8% | 97.4% |
| 50% (a coin) | 96.9% | 99.9% | ≈100% |
How to Read Your Result
Independence is an assumption
The formulas above assume one event does not change the chance of the other — true for coins and dice, not for drawing cards without putting them back, or for events with a shared cause. For dependent events you need conditional probabilities: P(A and B) = P(A) × P(B given A).
Rare events add up
A 1% chance is small once, but over many independent tries it becomes likely: about 63% after 100 tries. This is why rare risks matter at scale and why “it has never happened” is weak evidence for a low-probability event with few past tries.
Limitations & Accuracy Notes
- All events are assumed independent.
- Binomial mode supports up to 100,000 trials; results are computed with logarithms and are accurate to many significant digits.