When does the probability of an event equals one?

Answer 1

The probability of an event is one when the event is absolutely certain to happen.

It is more useful to understand why the sum of probabilities of an event adds up to one. If you have an event with a probability of p, then the probability of the event not occurring is 1 - p. This is because p plus (1 - p) is 1. That seems trivial, but consider a realistic example...

Lets say you are subject to random drug testing, and that you are in a 100% pool such as might be mandated by 10CFR26, "Fitness for Duty in a Commericial Nuclear Power Plant". What is the probability that you will be chosen at least once in a year?

Well, if sampling occurs five days a week, then there are 260 samples per year. The probability of being chosen in one sample is 1 in 260, or 0.003846. That means the probability of not being chosen is 259 in 260, or 0.9962. The probability, then, of not being chosen in one year is (259 in 260) raised to the 260th power, or 0.3672, which is about 95 in 260. Invert once again, and you see that the probability of being chosen at least one time in a year is 165 in 260, or 0.6328.