Page 18 - DMTH404_STATISTICS

Statistics

Notes The set operations like formation of intersection, union and complementation of two or more
sets that we have listed above and their combinations are sufficient for constructing new events
out of old ones. However, we need to express in a precise way commonly used expressions like
(i) if the event A has occurred, B could not have occurred and (ii) the occurrence of A implies that
of B. We’ll explain this by taking an example first.

Examples: Let us consider the following experiments.

(i) In the experiment of tossing a die twice, let A be the event that the total score is 8 and B that
the absolute difference of the two scores is 3. Then
A = | (x, y) | x + y = 8, x, y = 1, 2, 3, ..., 6 }

= { (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) }
and B = { (x,y) | | x – y | = 3, x, y = 1, 2, 3,..., 6 }
= { (1, 4) | | x – y | = 3, x, y = 1, 2, 3, ..., 6 }
(ii) Consider Experiment 11, where we select 13 cards without replacement from a pack of
cards. Let
event A : all the 13 cards are black and
event B : there are 6 diamonds and 7 hearts.

Note that in both the-cases there is no point which is common to both A and B. Or in other
words, A n B is the empty set. Therefore, in both i) and ii) we conclude that if A occurs, B
cannot occur and conversely, if B occurs A cannot occur. .
Now let us find an example for the sifuation : the occurrence of A implies that of B.
Take the experiment of tossing a die twice. Let A = { (x, y) | x + y = 12 } be the event that the
total score is 12, and B = { (x, y) | x – y = 0 } be the event of having the same score on both
the throws. Then
A= { (6,6) } and

B = { (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6);
so that whenever A occurs, B does. Note that A  B.
You were already familiar with the various operations on sets. In Sec. 5.4 we had . Sample
Space of a Random identified events with subsets of the sample space. What we have done
in. this section Experiment is to apply set operations to events, and to interpret the combined
events.
1.5 Summary

In this introductory unit to the study of probability, we have made the following points:
 There are many situations in real life as well as in scientific work which can be regarded
as experiments having more than one possible oatcome. We cannot predict the outcome
that we will obtain at the conclusion of the experiment. Such experiments are called
random experiments.
 The study of random experiments begins with a specification of its all possible outcomes.
In this specification, we have to make certain assumptions to avoid complexities. The set
of all possible outcome is called the sample space of the experiment. A sample space with
a finite number or a countable infinity of points is a discrete sample space. /

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