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Complementary Event

For every event A, there corresponds another event A' called the complementary event to A. It is also called the event 'not A'. For example, take the experiment 'of tossing three coins'. An associated sample space is


Let A = {HTH, HHT, THH} be the event 'only one tail appears'

Clearly for the outcome HTT, the event A has not occurred. But we may say that the event 'not A' has occurred. Thus, with every outcome which is not in A, we say that 'not A' occurs.

Thus the complementary event 'not A' to the event A is

or A' = {ω : ω S and ω A} = S - A.

The Event 'A or B'

Recall that union of two sets A and B denoted by A B contains all those elements which are either in A or in B or in both.

When the sets A and B are two events associated with a sample space, then 'A B' is the event 'either A or B or both'. This event 'A B' is also called 'A or B'.

Therefore Event 'A or B' = A B = {ω : ω A or ω B}

The Event 'A and B'

We know that intersection of two sets A B is the set of those elements which are common to both A and B i.e., which belong to both 'A and B'.

If A and B are two events, then the set A B denotes the event 'A and B'.

Thus, A B = {ω : ω A and ω B}

For example, in the experiment of 'throwing a die twice' Let A be the event 'score on the first throw is six' and B is the event 'sum of two scores is atleast 11' then

A = {(6,1), (6,2}, (6,3), (6,4), (6,5), (6,6)}, and B = {(5,6), (6,5), (6,6)}
so A B = {(6,5), (6,6)}

Note that the set A B = {(6,5), (6,6)} may represent the event 'the score on the first throw is six and the sum of the scores is atleast 11'

The Event 'A but not B'

We know that A - B is the set of all those elements which are in A but not in B.

Therefore, the set A - B may denote the event 'A but not B'. We know that
A - B = A B´.

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