Probability Calculator – Two Events and Repeated Trials
Combine the probabilities of two independent events, or find the chance of at least one, exactly k or at most k successes over repeated tries.
For independent events, P(A and B) = P(A) × P(B) and P(A or B) = P(A) + P(B) − P(A) × P(B). The chance of at least one success in n tries is 1 − (1 − p)ⁿ: with a 10 % chance each time, 10 tries give about 65 %.
How to use Probability Calculator
- Choose “Two events” or “Repeated trials”.
- Enter probabilities as decimals, percentages or fractions.
- For repeated trials, also enter the number of trials and successes.
- Read the combined probabilities.
- Check the note: the formulas assume independent events.
Basic rules
A probability is a number from 0 (impossible) to 1 (certain). 0.25, 25 % and 1/4 all mean the same thing.
- Not A: P(not A) = 1 − P(A)
- A and B (independent): P(A and B) = P(A) × P(B)
- A or B: P(A or B) = P(A) + P(B) − P(A and B)
Two events are independent when one does not change the chance of the other — two coin flips, or two dice. Drawing cards without putting them back is not independent, so these formulas do not apply there directly.
Example: two events
P(A) = 0.5 and P(B) = 0.3. Then P(A and B) = 0.15, P(A or B) = 0.65, P(neither) = 0.5 × 0.7 = 0.35 and P(exactly one) = 0.5.
At least one success
The easiest way to find “at least one” is through the opposite, “none”:
P(at least one) = 1 − (1 − p)ⁿ
| Chance per try | 5 tries | 10 tries | 20 tries |
|---|---|---|---|
| 1 % | 4.9 % | 9.6 % | 18.2 % |
| 10 % | 41.0 % | 65.1 % | 87.8 % |
| 25 % | 76.3 % | 94.4 % | 99.7 % |
Notice that ten tries at 10 % do not guarantee success; the chance is only about 65 %.
Exactly k successes: the binomial formula
P(exactly k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where C(n, k) is the number of ways to choose k of n. The probability of exactly 2 sixes in 10 dice rolls is C(10, 2) × (1/6)² × (5/6)⁸ ≈ 29.1 %. The calculator also adds up the cases for “at most k” and “at least k”, and shows the expected number of successes, n × p.
Common mistakes
- Adding probabilities of overlapping events without subtracting the overlap.
- Believing a run of bad luck makes success “due” — independent tries have no memory.
- Confusing odds with probability: odds of 1 to 3 mean a probability of 1/4.
For spread and averages of data, use the Standard Deviation Calculator calculator. To generate random picks, try the Password Generator, which uses the same secure randomness.
You may also find these free tools useful: GPA Calculator, LCM & GCF Calculator and Slope Calculator.
Frequently asked questions
How do you calculate the probability of A and B?
For independent events multiply them: P(A and B) = P(A) × P(B).
How do you calculate the probability of A or B?
Add them and subtract the overlap: P(A) + P(B) − P(A and B).
What is the probability of at least one success?
1 − (1 − p)ⁿ, where p is the chance per try and n the number of tries.
What is binomial probability?
The probability of exactly k successes in n independent tries with the same chance p each time.
Can I enter percentages or fractions?
Yes. Type 25%, 0.25 or 1/4 — they mean the same.
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