Mathematics · Statistics
Probability Calculator — Two Events, Binomial & At Least Once
Work out probabilities three ways: for two events — independent, mutually exclusive or with a known overlap — get P(A and B), P(A or B), neither, exactly one and P(A | B); for repeated trials get the binomial probability of exactly, at most or at least k successes; or find the chance of something happening at least once in n tries.
Your inputs
Quick start — try a typical case
As a decimal (0.5) — or a percentage above 1 (50).
Your result Live
P(A and B)
0.15 (15%)
0.5 × 0.3 — independent events multiply
- P(A or B)
- 0.65 (65%)
- P(neither)
- 0.35 (35%)
- P(exactly one)
- 0.5 (50%)
- P(A but not B)
- 0.35 (35%)
- P(A | B)
- 0.5 (50%)
- P(B | A)
- 0.3 (30%)
- P(not A) · P(not B)
- 0.5 · 0.7
How this is calculated
P(A∪B) = P(A) + P(B) − P(A∩B) · independent: P(A∩B) = P(A)P(B) · P(A|B) = P(A∩B) ÷ P(B) · binomial: C(n,k) pᵏ(1−p)ⁿ⁻ᵏ · at least once: 1 − (1 − p)ⁿ
- p
- Probability of success on one trial
For two independent events, the chance both happen is the product of their chances; for 'A or B' you add them and take away the overlap so it isn't counted twice. Repeated independent trials follow the binomial distribution. The chance of something happening at least once is easiest found backwards: one minus the chance it never happens.
Assumptions & sources
- Trials are independent with the same probability each time.
- 1Mathematics, Classes 11–12 — Probability and Statistics — NCERT
- 2NIST/SEMATECH e-Handbook of Statistical Methods — US National Institute of Standards and Technology
Calculator ID MATH-PROB-001 · version 1.0.0
Common questions
What's the chance of getting at least one six in 4 rolls?
1 − (5/6)⁴ ≈ 0.518 — just over 50 %.
What's the probability of exactly 3 heads in 10 tosses?
C(10,3) × 0.5¹⁰ = 120 ÷ 1024 ≈ 0.117.
What's the difference between independent and mutually exclusive?
Independent events don't affect each other (both can happen); mutually exclusive events can't happen together, so P(A and B) = 0.