Probability notebook
Why “at least one” never means “due”
The complement rule explains repeated chances without giving the next event a memory.
Start with the assumption
Suppose an event has a fixed probability p on each independent trial. The chance that it does not occur is 1 − p. Over n independent trials, the chance of no occurrence is (1 − p) raised to the power n. Subtract that result from one to find the chance of at least one occurrence. Our calculator implements that expression directly.
A worked example
With a hypothetical 10% chance per trial and ten independent trials, the chance of at least one occurrence is approximately 65.13%. It is neither 100% nor a prediction of the particular sequence you will observe. Multiplying ten by 10% answers a different question: the expected number of occurrences, which is one.
What the model leaves out
Independence and a constant probability are requirements, not decorative small print. A game with changing conditions needs a different model. A losing sequence also does not make a future independent event more likely. Use the experiment to understand the assumption, never to decide that a win must be imminent.