What is a random event in probability?
A concept of an event is an extremely important in the Theory of Probabilities. Actually, it's one of the fundamental concepts, like a point in Geometry or equation in Algebra.
First of all, we consider a random experiment - any physical or mental act that has certain number of outcomes. For example, we count money in our wallet or predict tomorrow's stock market index value. In both and many other cases the random experiment results in certain outcomes (the exact amount of money, the exact stock market index value etc.) These individual outcomes are called elementary events and all such elementary events associated with a particular random experiment together form a sample space of this experiment.
More rigorously, the sample space of any random experiment is a SET and all individual elementary events (that is, the individual results of this experiment) are ELEMENTS of this set.
Now we can consider not only an individual elementary event, like exact amount of money in a wallet, but a combination of such elementary events. For instance, we can consider the result of our money counting experiment to be less than $5. This is a combined event that consists of elementary events $0, $1, $2, $3 and $4. This and other combinations of elementary events is called a random event.
Using our SET terminology, a random event is a SUBSET of a SET of all elementary events (in other words, a SUBSET of a sample space). Any such SUBSET is called a random event.
In Theory of Probabilities there is a concept of probability associated with each elementary event. If the number of elementary events is finite or countable, this probability is just a non-negative number and the sum (even infinite sum in case of countable number of elementary events) equals to 1.
The probability associated with any random event is a sum of probabilities of all elementary events that comprise it.