How do you combine two independent probabilities?
Method: Multiply to find both events. For either event, add the individual probabilities and subtract the overlap. These rules apply only when the event relationship matches the assumption shown.
Combine independent event probabilities
- Convert each percentage to a decimal.
- Multiply the decimals to find both events.
- Subtract the overlap when calculating either event.
- Check that the four outcome groups sum to one.
Check the four outcome groups
If one event has probability 40% and another independent event has probability 30%, both occur with probability 12%. At least one occurs with probability 58%.
Independence is an assumption, not a result
Verify the relationship before multiplying
Two events are independent when learning one outcome does not change the chance of the other. Separate coin tosses may fit this model. Rain and umbrella use do not. When dependence is plausible, replace the second standalone probability with a probability conditional on the first event rather than using this calculator's shortcut.
Both, exactly one, and neither partition the possible outcomes. Their probabilities should sum to 100%. At least one overlaps several groups, so it is a summary rather than another separate outcome to add to that total.
Boundary of the result
Do not use the independence calculation when one event changes the chance of the other.
Continue with the method
Use the probability reasoning guide and the conditional probability article and the permutation and combination calculator to move from a quick result to a complete reasoning process.