If the two events are independent and neither event has only one outcome, the answer is no, because the number of outcomes is the product of the outcomes of each event taken separately. And the product of two numbers, each greater than 1, is a composite number, not a prime. For example, tossing a coin twice. This is a pair of binary events, one event taking place before the other. The outcomes are not related so there are four outcomes: HH, HT, TH, TT (heads and tails) including the order in which they occur (permutation, not merely combination) and this is illustrated by a tree diagram. There are 4 “leaves”.
Two events are separated by time and/or space. If the events can be linked by cause and effect (temporal events), so that they are not independent, one of the outcomes of the first event may preclude all but one outcome of the second event. In such a case, the total outcomes for the pair of events could be prime. For example, an astronaut can choose one of two spaceships, one bound for the Moon, the other to Mars. Each spaceship may malfunction at launch time. Or it may not. If it doesn’t malfunction, we can assume that it reaches its destination. The two events are:
- the choice that the astronaut makes; and
- whether the spaceship malfunctions.
So there are three (prime number) outcomes.
- The astronaut picks a spaceship that malfunctions, so he’s not going anywhere!
- The astronaut goes to the Moon.
- The astronaut goes to Mars.
So the answer in this case is yes, the number of outcomes can be prime.