7.3 Sample space
The totality of all the outcomes or results of a random experiment is denoted by Greek alphabet W or English alphabets and is called the sample space. Each outcome or element of this sample space is known as a sample print.
Event
Any subset of a sample space is called an event. A sample space S serves as the universal set for all questions related to an experiment 'S' and an event A w.r.t it is a set of all possible outcomes favorable to the even t A
For example,
A random experiment :- flipping a coin twice
Sample space :- W or S = {(HH), (HT), (TH), (TT)}
The question : "both the flipps show same face"
Therefore, the event A : { (HH), (TT) }
Equally Likely Events
All possible results of
a random experiment are called equally likely outcomes and we have
no reason to expect any one rather than the other. For example,
as the result of drawing a card from a well shuffled pack,
any card may appear in draw, so that the 52 cards become 52 different
events which are equally likely.
Mutually Exclusive Events
Events are called mutually exclusive or disjoint or incompatible if the occurrence of one of them precludes the occurrence of all the others. For example in tossing a coin, there are two mutually exclusive events viz turning up a head and turning up of a tail. Since both these events cannot happen simultaneously. But note that events are compatible if it is possible for them to happen simultaneously. For instance in rolling of two dice, the cases of the face marked 5 appearing on one dice and face 5 appearing on the other, are compatible.
Exhaustive Events
Events are exhaustive when they include all the possibilities associated with the same trial. In throwing a coin, the turning up of head and of a tail are exhaustive events assuming of course that the coin cannot rest on its edge.
Independent Events
Two events are said to be independent if the occurrence of any event does not affect the occurrence of the other event. For example in tossing of a coin, the events corresponding to the two successive tosses of it are independent. The flip of one penny does not affect in any way the flip of a nickel.
Dependent Events
If the occurrence or non-occurrence of any event affects the happening of the other, then the events are said to be dependent events. For example, in drawing a card from a pack of cards, let the event A be the occurrence of a king in the 1st draw and B be the occurrence of a king in the 1st draw and B be the occurrence of a king in the second draw. If the card drawn at the first trial is not replaced then events A and B are independent events.
Note
(1) If an event contains a single simple point i.e. it is a singleton set, then this event is called an elementary or a simple event.
(2) An event corresponding to the empty set is an "impossible event."
(3) An event corresponding to the entire sample space is called a ‘certain event’.
Complementary Events
Let S be the sample space for an experiment and A be an event in S. Then A is a subset of S. Hence  , the complement of A in S is also an event in S which contains the outcomes which are not favorable to the occurrence of A i.e. if A occurs, then the outcome of the experiment belongs to A, but if A does not occur, then the outcomes of the experiment belongs to
It is obvious that A and are mutually exclusive. A Ç = f and A È = S.
If S contains n equally likely, mutually exclusive and exhaustive points and A contains m out of these n points then contains (n - m) sample points.
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