statistics and probability
enter the chances of events a and b. choose independence or supply a known intersection to calculate the complete two-event breakdown.
assumes a and b are independent, so p(a and b) = p(a) × p(b).
enter event probabilities and calculate.
conditional p(a given b): —. this is undefined when p(b) = 0.
p(a and b) is the chance that both events occur. if a and b are independent, multiply their probabilities: p(a and b) = p(a) × p(b). independence is an assumption about the underlying situation, not something this calculator can infer from two percentages alone.
p(a or b) includes outcomes where either event occurs, including the overlap. the addition rule is p(a or b) = p(a) + p(b) - p(a and b). subtracting the intersection prevents double counting. 'exactly one' removes the overlap entirely.
for related events, provide p(a and b) directly from data or a valid model. the intersection cannot exceed the smaller event probability and cannot be less than p(a) + p(b) - 100% when the events together exceed 100%. these bounds are checked so all four regions of the two-event diagram remain nonnegative.
conditional probability answers a different question: p(a given b) = p(a and b) / p(b), provided p(b) is not zero. when p(b) = 0 the conditional value is undefined, not 0%. all results describe the same experiment and sample space; they do not validate whether your input estimates are representative.
for independent events with p(a) = 40% and p(b) = 30%, both = 0.4 × 0.3 = 12%. either = 40% + 30% - 12% = 58%; neither = 42%.
a only = p(a) - p(a and b); b only = p(b) - p(a and b); exactly one = a only + b only.
for dependent events with p(a) = 70%, p(b) = 60% and p(a and b) = 40%, either = 90%, neither = 10% and p(a given b) = 40 / 60 = 66.67%.
model the chance of two inspection criteria when their individual and joint frequencies are known.
compare two defined events only when their probabilities refer to the same place, period and sample space.
see why the intersection must be subtracted once from p(a) + p(b).
a formula cannot correct biased observations or justify an independence assumption; verify input definitions and source data before making decisions.
add p(a) and p(b), then subtract p(a and b) so the overlap is counted only once.
select known intersection and enter p(a and b) from a valid source. do not multiply p(a) and p(b) unless independence is justified.
it must lie between max(0, p(a)+p(b)-100%) and min(p(a),p(b)); outside this range the event probabilities cannot all be true.
conditional probability divides by p(b), so it has no defined value when p(b) is zero.
yes. in the standard probability addition rule, a or b is inclusive and includes outcomes where both occur.
last reviewed: october 23, 2026. calculations and simulations run in your browser; verify important decisions independently.
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