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At least one

The probability of at least one success is 1 minus the probability of no successes. That is often shorter than adding one, two and more. When the trials are independent and have the same probability of failure, "none" is that probability raised to the number of trials. The answer lies between 0 and 1. The more trials, the greater the chance of at least one.

P(\text{at least one}) = 1 - P(\text{none})

the complement rule

P(\text{none}) = (1 - p)^n

independent trials

Symbols

$p$	probability of a hit in one trial
$n$	number of trials

Example

Each day the probability of dry weather is 0.8, independently of the other days.

Over three days, $P(\text{at least one rainy day}) = 1 - 0.8^3 = 0.488$ .

"At least one" is the complement of "none".

Practise probability for free →

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Part of Mathematics S1: Probability.