Page 24 - DMTH404_STATISTICS

Statistics

Notes Definition 1 : The probability P(.4) of an event A is the sum of the Probabilities of the points
in A. More formally,


P(A)   P{ }....(2)
j
j   A
where  stands for the fact that the sum is taken over all the points   A, A is, of course, a
j
j   A
subset of . By convention, we assign probability zero to the empty set. Thus, P( ) = 0.
j
The following example should help in clarifying this concept.

Example 2: Let a be the sample space corresponding to three tosses of a coin with the
following assignment of probabilities.
Sample point HHH HHT HTH THH TTH THT HTT TIT
Probability 1/8 1/8 118 1/8 1/8 1/8 1/8 1/8
Let’s find the probabilities of the events A and B, where

A : There is exactly one head in three tosses, and
B : All the three tosses yield the same result
Now A = ( HTF, THT, TITH )
Therefore,

P(A) = 1/8 + 1/8 + 1/8 = 318.
1 1 1
Further, B = {HHH, TTT). Therefore, P(B) =   .
8 8 4
Proceeding along these lines you should be able to do this exercise.
A word about our notation and nomenclature is necessary at this stage. Although we say that
P{} is the probability assigned to the point wj of the sample space, it can be also interpreted as
j
the probability of the singleton event {}.
j
In fact, it would be useful to remember that probabilities are defined only for events and that
P{ } is the probability of the singleton event { }. This type of distinction will be all the
j j
more necessary when you proceed to study probability theory for non-discrete sample spaces in
Block 3.
Now let us look at some. of the probabilities of events.

2.1.2 ProbabJlity of an Event : Properties

By now you know that the probability P(A) of an event A associated with a discrete sample space
is the sum of the probabilities assigned to the sample points in A. In this section we discuss the
properties of the probabilities of events.
P1: For every event A, 0  P(A)  1.
Proof: This is a straightforward consequence of the definition of P(A). Since it is the sum of
non-negative numbers, P(A)  0. Since the sum of the probabilities assigned to all the points in
the sample space is one and since A is a subset of R, the sum of the probabilities assigned to
the points in A cannot exceed P(R), which is one. In other words, whatever may be the event
A, 0  P(A)  1.

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