MultiConvers

Probability Calculator

Apply one of five rules to percentages you already have: independent “and”, exclusive “or”, overlapping “or”, the complement, or “at least once” in n independent trials. The page does not invent dependence or update a prior.

How to use

  1. Pick the rule that matches the story. Independent “and” multiplies; exclusive “or” adds; overlapping “or” needs a known P(A and B).
  2. Enter probabilities as percentages — 30 means 30%, not 0.30. Values outside 0–100 are rejected.
  3. For “at least once”, also enter how many independent trials n (a positive integer).
  4. Read the result as a percentage. Switch rules instead of forcing the wrong identity onto the same pair of events.

How it works

This calculator only evaluates the five identities in the formula box. Independent P(A and B) is the product: 30% and 40% become 12%. That product is legal only when knowing A happened does not change B. Rain and a coin toss can be treated as independent; “has a fever” and “tested positive” usually cannot. If you multiply dependent events you under- or over-count, and the page has no data with which to notice.

Mutually exclusive “or” adds the two percentages and refuses a sum above 100%. Exclusive means they cannot both occur — drawing a heart and drawing a spade in one card. Overlapping “or” is the inclusion-exclusion version: P(A) + P(B) − P(A and B). You must supply the overlap. The overlap cannot exceed either event; “30% and 40% with 35% both” is rejected. Forgetting to subtract the overlap is how people claim a 70% chance of “rain or wind” when those two already share a stormy 20%.

The complement is 1 − P(A). A 25% chance of A is a 75% chance of not A. That is the cleanest of the five rules and the usual way to flip a “success” into a “failure” before the last mode. “At least once” in n independent trials is 1 − (1 − P(A))ⁿ — the chance of avoiding A every time, subtracted from one. It is the standard shortcut for “will I see at least one six in four rolls?” when each trial has the same P(A) and does not change the others. n must be a positive integer.

What the tool does not compute: Bayes’ rule, conditional probability from a test’s sensitivity, a binomial coefficient times p^k, or a z-table tail. It will not take a count of permutations and turn it into a probability. Inputs must lie in 0–100. Exclusive “or” that would exceed 100%, or an overlap larger than A or B, fails rather than wrapping around.

Use permutations and combinations when you are still counting equally likely outcomes. Use this page when those counts (or a published rate) have already become percentages and you only need to combine them under a named assumption.

Formula

P(A and B) = P(A)P(B)   (independent)
P(A or B) = P(A) + P(B)   (exclusive)
P(A or B) = P(A) + P(B) − P(A and B)
P(not A) = 1 − P(A)
P(at least one) = 1 − (1 − P(A))ⁿ

Worked examples

  • Independent and

    30% and 40% → 12%

  • Complement

    P(A) = 25% → P(not A) = 75%

Useful notes

  • This is not a Bayes calculator and does not update from a test result.
  • Enter 30 for 30%. The fields are percentages, not decimals.
  • Exclusive “or” cannot exceed 100%, and a stated overlap cannot exceed either event.

FAQ

When can I multiply probabilities?

When the events are independent — knowing one happened does not change the other.

When can I add them?

When they cannot happen together (mutually exclusive). Otherwise subtract the overlap.

Why do I enter 30 instead of 0.3?

These fields are percentages. 30 means 30%.

What does “at least once in n trials” assume?

The same probability P(A) on every independent trial. The formula is 1 − (1 − P(A))ⁿ. It is not a binomial probability for exactly k successes.

Can I use this for a medical test’s positive predictive value?

No. That needs Bayes’ rule and a prevalence. This page only combines percentages under the five listed rules.

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