91Ó°ÊÓ

The sample space S of some experiment is countable. For every outcome \(s \in S\), the subset \(\left\{ s \right\}\) is an event.

When we consider an experiment of tossing two coins simultaneously, the 4 possible outcomes are: HH, HT, TH, and TT.

Hence, the sample space S consisting of all outcomes is \(S = \left\{ {HH,HT,TH,TT} \right\}\).

The sample space can also be written as,\(S = \left\{ {{s_1},{s_2},{s_3},{s_4}} \right\}\), where \({s_i}\left( {\forall i = 1,2,3,4} \right)\) denotes the ith outcome in S.

It is already given that \(\left\{ {{s_i}} \right\}\) is an event for all \(i = 1,2,3,4\).

So, \(\left\{ {HH} \right\},\left\{ {HT} \right\},\left\{ {TH} \right\},\left\{ {TT} \right\}\)are all events.

Considering the other subsets of S taking 2, 3, and 4 outcomes at a time, let\({S_2} = \left\{ {{s_i},{s_j}} \right\};\left( {\forall i,j \in \left\{ {1,2,3,4} \right\};i \ne j} \right)\), where \({S_2}\) is a subset of S that takes any two outcomes of S at a time.

Say, \({S_2} = \left\{ {HH,TH} \right\}\). This expression denotes that either both tosses result in heads or the first toss results in tail and the second one in head. Since this subset has two well defined outcomes, it is also considered an event.

Similarly, every other subset of S has well defined outcomes in them, and hence, can be considered an event.

Thus, every subset of Sis an event.