Probability Calculator (Events & Binomial)

Events

Enter probabilities as decimals (0.25) or percentages (25).

P(A)Decimal 0–1 or percent 0–100

P(B)

Relationship between A and B

P(A and B)Used only in the third mode

Binomial: number of trials (n)

Binomial: successes (k)

P(A OR B)

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P(A and B)—

P(A | B) — A given B—

P(B | A) — B given A—

P(not A)—

P(neither A nor B)—

P(exactly one of them)—

Odds for A—

Consistency check—

Binomial — n trials at P(A)

P(exactly k successes)—

P(at most k)—

P(at least k)—

P(at least one)—

Expected successes—

Standard deviation—

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Independent is not the same as mutually exclusive

These two get confused constantly and they are close to opposites. Independent means knowing one happened tells you nothing about the other, so P(A and B) = P(A) × P(B). Mutually exclusive means they cannot both happen, so P(A and B) = 0 — and that makes them maximally dependent, because learning A happened tells you B definitely did not.

Rolling a 6 and drawing an ace are independent. Rolling a 6 and rolling a 3 on the same die are mutually exclusive. Treating the second pair as independent would give 1/6 × 1/6 for something that can never occur.

The "at least one" trick

Almost every "what are the chances of at least one" question is easier answered backwards. P(at least one) = 1 − P(none). For n independent attempts each with probability p, that is 1 − (1 − p)ⁿ. Ten attempts at 10% each gives 65%, not 100% — a result that catches people out because the naive sum of probabilities happens to reach exactly 1.

Conditional probability and the base rate

P(A | B) = P(A and B) / P(B). The denominator is the trap: it restricts the world to cases where B happened, which can change the answer enormously when B is rare. This is why a test that is "99% accurate" for a disease affecting 1 in 10,000 people still produces mostly false positives — the base rate dominates, and the conditional formula is where that becomes visible.

Binomial: repeated identical trials

The binomial block assumes n independent trials, each with the same success probability, each with only two outcomes. P(exactly k) = C(n,k) p^k (1−p)^n−k. Break any of those assumptions — trials that influence each other, a probability that drifts — and the model quietly stops applying even though it still returns a number.

Frequently Asked Questions

How do I find the probability of A or B?

Add them and subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B). Forgetting to subtract the overlap double-counts the cases where both happen, which is the most common error in this calculation.

What is the probability of at least one success in n tries?

One minus the probability of no successes: 1 − (1 − p)ⁿ. The binomial block above reports this directly. It rises fast at first and then flattens, never quite reaching certainty.

Can a probability be greater than 1?

No. If your inputs produce a value above 1, the numbers are inconsistent — usually P(A and B) has been set larger than P(A) or P(B), which is impossible. The consistency check field flags this.

What does the binomial standard deviation tell me?

It is √(np(1−p)) and describes how much the number of successes typically varies between runs. With n = 100 and p = 0.5 the expected count is 50 with an SD of 5, so results between 40 and 60 are entirely ordinary.

Where these numbers come from

Sources

Official publications only. Links open the original document in a new tab.

National Institute of Standards and Technology Special Publication 811 — Guide for the use of the SI Significant figures and rounding conventions
National Institute of Standards and Technology CODATA fundamental physical constants Values of the fundamental physical constants

Probability Calculator