Probability Calculator

Related Calculators

Event relationship / what you know: The rule the calculator should use for A and B. Pick independent, mutually exclusive, or general overlap known.

Probability of event A: The chance that event A happens, entered as a decimal from 0 to 1.

Probability of event B: The chance that event B happens, entered as a decimal from 0 to 1.

Probability of both A and B, P(A and B): The overlap, meaning the chance that A and B happen together.

Probability of A or B: The chance that at least one of the two events happens. It includes the overlap unless the events are mutually exclusive [2].

Probability of A only: The chance that A happens and B does not.

Probability of exactly one event: The chance that one event happens but not both.

Probability of neither A nor B: The chance that neither event happens.

Conditional probability P(A given B): The chance that A happens after you already know B happened [1].

Mutually exclusive events: Events that cannot happen together, so they have no outcomes in common [2].

Independent events: Events where knowing one happened does not change the probability of the other [2].

Mistake: Typing 50 into "Probability of event A" when the label expects a decimal probability.
Fix: Enter 0.50 for 50%, or switch only the "Display format" if you want the answer shown as a percent.

Mistake: Choosing "Independent events" when the problem really says the events cannot happen together.
Fix: Change "Event relationship / what you know" to "Mutually exclusive events" so "Probability of A and B" becomes 0.

Mistake: Entering a "Probability of both A and B, P(A and B)" value that is bigger than "Probability of event A" or "Probability of event B".
Fix: In the general case, make sure the overlap is less than or equal to both single-event probabilities.

Mistake: Adding probabilities to answer an "and" question.
Fix: Check the output labels carefully: use "Probability of A and B" for both happening, and use "Probability of A or B" for at least one happening.

Mistake: Forgetting that "Conditional probability P(A given B)" depends on B already having happened.
Fix: Read it as "among cases where B happened, how often does A happen" and compare it with "Probability of event A" only if that wording matches your question.

Mistake: Treating a rounded fraction display as if the calculator changed the actual math.
Fix: The calculator keeps decimal math internally. If a fraction looks too simple, increase "Decimal places" and compare with Decimal display.

• All entered probabilities must be between 0 and 1 inclusive. Values outside that range are invalid.
• This calculator handles one or two events only. It is not a binomial, Bayes, normal distribution, or multi-event tool.
• In "Independent events" mode, the calculator assumes independence because you selected it. If that assumption is wrong, the overlap and conditional results will be wrong.
• In "Mutually exclusive events" mode, the calculator assumes the events cannot happen together, so "Probability of A and B" is forced to 0.
• In "General case: I know P(A and B)", the overlap must be less than or equal to both single-event probabilities, and the implied "Probability of A or B" cannot exceed 1.
• Conditional outputs are undefined when the condition has probability 0. In that case, the calculator shows N/A instead of dividing by zero.
• Fraction output is a display approximation with denominator up to 1000. The calculator still uses decimal values internally, so some repeating decimals may be shown as nearby simple fractions.
• Rounded display values may not add perfectly due to rounding, even when the full internal results are consistent.

How the calculator chooses the overlap

The first choice is the relationship between the events.

Independent mode: P(A and B) = P(A) * P(B)

Mutually exclusive mode: P(A and B) = 0

General overlap known mode: P(A and B) = entered overlap

Independent means one event does not change the probability of the other [2]. Mutually exclusive means the events cannot both happen and have no outcomes in common [2].

Core formulas

After the overlap is known, the calculator finds the other outputs from standard two-event rules.

P(not A) = 1 - P(A)

P(not B) = 1 - P(B)

P(A or B) = P(A) + P(B) - P(A and B)

P(A only) = P(A) - P(A and B)

P(B only) = P(B) - P(A and B)

P(exactly one) = P(A only) + P(B only)

P(exactly one) = P(A) + P(B) - 2*P(A and B)

P(neither) = 1 - P(A or B)

P(A given B) = P(A and B) / P(B)

P(B given A) = P(A and B) / P(A)

If P(B) = 0, then P(A given B) is undefined, so the calculator shows N/A instead of dividing by zero. If P(A) = 0, it does the same for P(B given A).

Worked mini-example

Suppose P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.2 in the general case.

P(A or B) = 0.6 + 0.5 - 0.2 = 0.9

P(A only) = 0.6 - 0.2 = 0.4

P(B only) = 0.5 - 0.2 = 0.3

P(exactly one) = 0.4 + 0.3 = 0.7

P(neither) = 1 - 0.9 = 0.1

P(A given B) = 0.2 / 0.5 = 0.4

P(B given A) = 0.2 / 0.6 = 0.3333...

This means at least one of the events happens 90% of the time, exactly one happens 70% of the time, and neither happens 10% of the time.

Validation logic

The calculator checks for impossible inputs before showing results.

0 <= each entered probability <= 1

Mutually exclusive mode: P(A) + P(B) <= 1

General mode: P(A and B) <= P(A)

General mode: P(A and B) <= P(B)

General mode: P(A) + P(B) - P(A and B) <= 1

If one of these checks fails, the calculator blocks the result and explains which input combination is impossible.

Display method

All math is done with decimal probabilities. Percent output is just decimal times 100. Fraction output is only a display approximation using a denominator up to 1000, so a displayed fraction may be rounded even when the internal decimal result is more precise.