We have seen the example of the uncertain event a “Spurs win the FA Cup in the year 2021”. We can think of this event as just one state of the variable A which represents “FA Cup winner in 2021”. In this case A has many states, one for each team entering the FA Cup. We write this as A = {a1, a2, …, an}, where a1 = “Spurs”, a2 = “Chelsea”, a3 = “West Ham”, etc. Since in this case the set A is finite we say that A is a finite discrete variable.

As another example, suppose we are interested in the number of critical faults in our control system. The uncertain event is A = “Number of critical faults”. Again it is best to think of A as a variable which can take on any of the discrete values 0,1,2,3,… thus A = {0,1,2,3,…}. Let us define a1 as the event “A=0″, and a2 as the event “A=1″. Clearly the events a1 and a2 are mutually exclusive and so P(a1 or a2)=P(a1)+P(a2). However, we cannot say that P(a1 or a2) = 1, because a1 and a2 are not exhaustive. That is, they do not form a complete partition of A. However, if we define a3 as the event “A>1″ then a1, a2, and a3 are complete and mutually exhaustive and in this case P(a1)+P(a2)+P(a3) = 1.

In general if A is a variable with states a1, a2, …, an:

The probability distribution of A, written P(A), is simply the set of values {P(a1), P(a2), …, P(an)}.