Page 14 - DMTH404_STATISTICS

Statistics

Notes
Examples: Suppose we toss a coin twice. The sample space of this experiment is
= (HH, HT, TH, TT}, where HT stands for a head followed by a tail, and other points are
similarly defined. Let’s list all the events associated with this experiment. There are 16 such
events. These are :
,{HH}, {HT}, {TH}, {TT}
{HH, HT}, {HH, TH}, {HH, TT}, {HT, TH}
{HH, TT}, {TH, TT}, {HH, HT, TH}, {HH, TH, TT},

{HH, HT, TT}, {HT, TH, TT}, 
Since we have identified an event with a subset of , the class of all events is the class of all the
 N 
subsets of . If  has N points, for a fixed r, we can form   sets consisting of r points, where
 r 
r = 0, 1, ... , N. The total number of events is, thirefore,

 N   N   N  N N


     ...     (1 1)  2 .
 0   1   N 

Notes By binomial theorem

 N   N   N 
N
N
(1 x)      x ...   x .



 0   1   N 
4
10
In Example 1, N = 4. Therefore, we have 2 = 16 events. If N = 10, we shall 2 = 1024 events. The
number of events thus increases rapidly with N. It is infinite if the sample space is infinite.
Let us now clarify the meaning of the phrase “The event A has occurred.”
We continue with Experiment 5. Let A denote the event { ,  ,  ) = {BBG, BGB, GBB}. If, after
5 5 7
performing the experiment, our outcome is  = BBG, which is a point of the set A, we say that
5
the event A has occurred. If, on the other hand, the outcome is  = BBB, which is not a point of
8
A, then we say that A has not occurred. In other words, given the outcome  of the experiment,
we say that A has occurred if   A and that A has not occurred if   A.
On the other hand, if we only know that A has occured, all we know is that the outcome of the
experiment is one of the points of A. It A then not possible to decide which individual outcome
has resulted unless A is a singleton.
In the next section we shall talk about some ways of combining events.

1.4 Algebra of Events

In this section we shall study different ways in which we can combine two or more events. We
shall also study relations ktween them. Since we are dealing with discrete sample spaces and
since any subset of the sample space is an event, we shall use the terms event and subset
interchangeable.
In what follows, events and sets are denoted by capital letters A, B, C, ... , with or without
suffixes. We shall assume that they all consist of points chosen from the same sample space .

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