Since ((HT)n((HH)=8, P({(HT)}U{(HH)}=P({(HT)}+P(0{(HH)} 111 42 To summarize the argument so far, Kolmogorov formalized the condition(a) and () of the random experiment in the form of the trinity(S, F, P()com- prising the set of all outcomes S-the sample space a a-field F of events re- ated to S and a probability function P() assigning probability to events in F For the coin example, if we choose F(The first is H and the second is T) &(HT), I (TH), (HH), (TT)1,8, S to be the o-field of interest, P() is defined PS)=1,P()=0.P({(HD))=4,((TH),(Hm,(T)})=4 Because of its importance the trinity(S, F, P() is given a name Definition 6: A sample space S endowed with a a-field F and a probability measure P(is called a probability space. That is we call the triple (S, F, p) a probability As far as condition()of 8 is concerned, yet to be formalized, it will prove of paramount importance in the context of the limit theorems in Chapter 4 2.2 Conditional Probability So far we have considered probabilities of events on the assumption that no information is available relating to the outcome of a particular trial. Sometimes however, additional information is available in the form of the known occurrence of some event A. For example, in the case of tossing a fair coin twice we might know that in the first trial it was heads. what difference does this information make to the original triple(S, F, p)? Firstly, knowing that the first trial was a head, the set of all possible outcomes now becomes SA={(H),(HH)}Since {(HT)} ∩ {(HH)} = ∅, P({(HT)} ∪ {(HH)}) = P({(HT)}) + P(∩{(HH)}) = 1 4 + 1 4 = 1 2 . To summarize the argument so far, Kolmogorov formalized the condition (a) and (b) of the random experiment E in the form of the trinity (S, F,P(·)) com￾prising the set of all outcomes S–the sample space, a σ-field F of events re￾lated to S and a probability function P(·) assigning probability to events in F. For the coin example, if we choose F(The first is H and the second is T)= {{(HT)}, {(T H),(HH),(TT)}, ∅, S} to be the σ-field of interest, P(·) is defined by P(S) = 1, P(∅) = 0, P({(HT)}) = 1 4 , P({(T H),(HH),(TT)}) = 3 4 . Because of its importance the trinity (S, F,P(·)) is given a name. Definition 6: A sample space S endowed with a σ-field F and a probability measure P(·) is called a probability space. That is we call the triple (S, F,P) a probability space. As far as condition (c) of E is concerned, yet to be formalized, it will prove of paramount importance in the context of the limit theorems in Chapter 4. 2.2 Conditional Probability So far we have considered probabilities of events on the assumption that no information is available relating to the outcome of a particular trial. Sometimes, however, additional information is available in the form of the known occurrence of some event A. For example, in the case of tossing a fair coin twice we might know that in the first trial it was heads. What difference does this information make to the original triple (S, F,P) ? Firstly, knowing that the first trial was a head, the set of all possible outcomes now becomes SA = {(HT),(HH)}, 10