Sets, Events and Probability
Suppose that we toss a coin any number of times and that we list the information of whether we got heads or tails on the successive tosses:
H T H H H T H T T T T T T H H · · · 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 · · ·
The act of tossing this coin over and over again as we have done is an example of an experiment, and the sequence of H’s and T’s that we have listed is called its experimental outcome. We now ask what it means, in the context of our experiment, to say that we got a head on the fourth toss. We call this an event. While it is intuitively obvious what an event represents, we want to find a precise meaning for this concept. One way to do this which is both
obvious and subtle is to identify an event with the set of all ways that the event can occur. For example, the event “the fourth toss is a head” is the same as the set of all sequences of H’s and T’s whose fourth entry is H. At first it appears that we have said little, but in fact we have made a conceptual advance. We have made the intuitive notion of an event into a concrete notion: events are a certain kind of set.