Page 197 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes
Example: of events are:
1. The event that the sum of two dice is 10 or more,

A = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}.
2. The event that a machine lives less than 100 days,
A = (0, 100) .
3. The event that out of fifty chosen people, five are left-handed,
A = {5} .

Example: (Coin Tossing) Presume that a coin is tossed 3 times, and that we “record”
every head and tail. The sample space can then be shown as
 = {HHH,HHT,HTH,HTT, THH, THT, TTH, TTT} ,
where, for instance, HTH means that the first toss is heads, the second tails, and the third heads.
An substitute sample space is the set {0, 1}3 of binary vectors of length 3, e.g., HTH communicate
to (1,0,1), and THH to (0,1,1). The event A that the third toss is heads is
A = {HHH,HTH,THH,TTH} .

As events are sets, we can affect the common set operations to them:
1. The set A  B (A union B) is the event that A or B or both take place,
2. The set A  B (A intersection B) is the event that A and B both take place,
3. The event Ac (A complement) is the event that A does not crop up,
4. If A  B (A is a subset of B) then event A is said to imply event B.
Two events A and B which have no results in general, that is, AB =  , are known as disjoint events.

Example: Let us cast two dice successively. The sample space is  = {(1, 1), (1, 2), ...,
(1, 6), (2, 1),..., (6, 6)}. Let A = {(6, 1), ..., (6, 6)} be the event that the first die is 6, and let B =
{(1, 6), ..., (1, 6)} be the event that the second dice is 6. Then A B = {(6, 1), . . . , (6, 6)}{(1, 6), ...,
(6, 6)} = {(6, 6)} is the event that both die are 6.

It is frequently functional to portray events in a Venn diagram, like in Figure 14.7.

Figure 14.7: A Venn Diagram

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