Events

 Events 

Another basic notion is the concept of an event. An event A (with respect to a particular sample space S associated with an experiment 8) is simply a set of possible outcomes. In set terminology, an event is a subset of the sample space S. In view of our previous discussion, this means that S itself is an event and so is the empty set 0. Any individual outcome may also be viewed as an event. The following are some examples of events. Again, we refer to the above-listed experiments: A_i will refer to an event associated with the experiment E_i.
A₁ : An even number occurs; that is, A₁ = {2, 4, 6}.
A₂: {2}; that is, two heads occur.
A₃: {HHHH, HHHT, HHTH, HTHH, THHH}; that is, more heads than tails showed.
A₄: {O} ; that is, all items were non defective.
A₅: {3, 4, . . . , M} ; that is, more than two rivets were defective.
A₆: {t | t < 3}; that is, the bulb burns less than three hours.
A₁₄: {(x, y) | y = x + 20}; that is, the maximum is 20° greater than the minimum.

When the sample space S is finite or countably infinite, every subset may be considered as an event. [It is an easy exercise to show, and we shall do it shortly, that if S has n members, there are exactly 2 n subsets (events).] However, if S is non countably infinite, a theoretical difficulty arises. It turns out that not every conceivable subset may be considered as an event. Certain "non admissible" subsets must be excluded for reasons which are beyond the level of this presentation. Fortunately such non admissible: sets do not really arise in applications and hence will not concern us here. In all that follows it will be tacitly assumed that whenever we speak of an event it will be of the kind we are allowed to consider. We can now use the various methods of combining sets (that is, events) and obtain the new sets (that is, events) which we introduced earlier.
(a) If A and Bare events, A ∪ B is the event which occurs if and only if A or B (or both) occur.
(b) If A and B are events, A ∩ B is the event which occurs if and only if A and B occur.
(c) If A is an event, A is the event which occurs if and only if A does not occur.
(d) If A i. . . . , An is any finite collection of events, then \[\bigcup\limits_{i=1}^{n}{{{A}_{i}}}\] is the event which occurs if and only if at least one of the events Ai occurs.
(e) If A₁  . .. , A_n is any finite collection of events, then \[\bigcap\limits_{i=1}^{n}{{{A}_{i}}}\] is the event which occurs if and only if all the events A_i occur.
(f ) If A₁. ... , A_n, ... is any (countably) infinite collection of events, then \[\bigcup\limits_{i=1}^{∞}{{{A}_{i}}}\] is the event which occurs if and only if at least one of the events A; occurs.
(g) If A i. .. . An, . . . is any (countably) infinite collection of events, then \[\bigcap\limits_{i=1}^{∞}{{{A}_{i}}}\] is the event which occurs if and only if all the events A_i occur.
(h) Suppose that S represents the sample space associated with some experiment ε and we perform ε twice. Then S x S may be used to represent all outcomes of these two repetitions . .That is, (s₁. s₂) ∈ S x S means that s₁ resulted when S was performed the first time and s₂ when ε was performed the second time.
(i) The example in (h) may obviously be generalized. Consider n repetitions of an experimerrt ε whose sample space is S. Then S x S x · · · x S = {(s_i. s₂, ••• , s_n), s_i ∈ S, i = 1, ... , n} represents the set of all possible outcomes when 8 is performed n times. In a sense, S x S x · · · x S is a sample space itself, namely the sample space associated with n repetitions of S.

Definition

Two events, A and B, are said to be mutually exclusive if they cannot occur together. We express this by writing A ∩ B = ∅; that is, the intersection of A and B is the empty set.

EXAMPLE 1

An electronic device is tested and its total time of service, say t, is recorded. We shall assume the sample space to be {t | t ≥ O}. Let the three events A, B, and C be defined as follows:
A = {t | t < 100}; B = {t | 50 ≤ t ≤ 200}; c = {t | t > 150}.
Then
A ∪ B = {t | t ≤ 200}; A ∩ B = {t | 50 ≤ t < 100};
B ∪ c = {t | t ≥ 50}; B ∩ c = {t | 150 < t ≤ 200};A ∩ c = ∅;
A ∪ C = {t | t < 100 or t > 150};  \[\bar{A}\] = {t | t ≥ 100}; \[\bar{C}\] = {t |  t ≤ 150}.

One of the basic characteristics of the concept of "experiment" as discussed in the previous section is that we do not know which particular outcome will occur when the experiment is performed. Saying it differently, if A is an event associated with the experiment, then we cannot state with certainty that A will or will not occur. Hence it becomes very important to try to associate a number with the event A which will measure, in some sense, how likely it is that the event A occurs. This task leads us to the theory of probability.