Suppose that the sample space of S of some experiment is cou

Suppose that the sample space of S of some experiment is countable. Suppose also that, for every outcome s, the subset {s} is an event. Show that every subset of S must be an event.

Solution

Lets prove it with an example.

Let sample space S be : {1,2,3,4,5}

Given that for every outcome s, the subset {s} is an event.

Therefore,

Occurance of 1 is an event.

Occurance of 2 is an event

Occurance of 3 is an event

Occurance of 4 is an event

Occurance of 5 is an event.

Now , if we consider two outcome taken together:

Say {1,2} , it is also an event when the outcome is a pair {1,2} and the associated probability would be 1/(5C2).

Similarly,

Say {1,2,3} , it is also an event when the outcome is a triplet {1,2,3} and the associated probability would be 1/(5C3).

and so on..... till {1,2,3,4,5}.

Hence , every subset of S must be an event. Proved