Find the chance all or at least one of several independent events occurs. A checkable formula accompanies combined probabilities instead of leaving an unexplained number.
For probabilities 0.8,0.65,0.9, all three occur with probability 0.468. At least one occurs with probability 0.993.
Independent events do not change one another’s probabilities. Their joint probability is the product of individual probabilities, while at least one is one minus the product of all complements.
The calculation treats event probabilities according to their labels, not as interchangeable values in independent events.
For all events, multiply probabilities. For at least one, multiply their failure probabilities and subtract the result from one. Independent Events also leads to identical trials.
Independence is a modeling assumption, not something the arithmetic proves. Shared causes or sampling without replacement can invalidate multiplication. If the Independent Events assumptions do not fit, consider nonoverlapping events.
Save extra digits internally if combined probabilities will become an input to another calculation.
The formulas combine repeated component reliability, independent test results, separate game trials, and multi-stage success conditions.
A field name is part of the formula. Match the problem's quantities to event probabilities, then check that they share the scale assumed by the independent events relationship.
Adding another noncertain requirement lowers the chance all occur but raises the chance at least one occurs. This is a stronger check than judging the answer only by how many decimal places it shows.
Match the reported precision to event probabilities, not to the number of digits the browser can display. Preserve an exact form when it communicates the independent events structure more clearly than a decimal.
Mutually exclusive events cannot occur together; independent nontrivial events can. The formula panel makes the chosen definition explicit.
The minimum audit trail is short: event probabilities, their units, and the formula beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
Before carrying a Independent Events result into later work, vary one of Event probabilities and predict whether the answer should rise, fall, or remain unchanged. This sensitivity check is especially useful when similar fields represent totals, successes, intersections, or conditioning events.
For Independent Events, a surprising response should trigger a model review before a rounding change. Confirm independence, replacement, event orientation, and row numbering in the source problem, because arithmetic cannot repair a sample space defined incorrectly.
No.
Multiply all complements.
Yes.
Usually not.