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# Algebra of Events

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#### Algebra of Events - Lesson Summary

Consider a sample space S, and two events of S, A and B. The basic algebra of events includes complementary event, event "A or B", event "A and B", and event "A but not B".

Complementary events
The complement of an event A is the event consisting of all the outcomes of the sample space that do not correspond to the occurrence of A. It is also called event "not A".

The complement of A is denoted by A' or  A _ .

Consider the sample space, S = {1, 2, 3, 4, 5, 6}

A = {3, 6} is an event. For outcome 1, event A has not occurred i.e. "not A" has occurred.

Similarly, for every outcome that is not in A, "not A" has occurred.

Thus, the complementary event "not A" to event A, or A' is {1, 2, 4, 5}.

A' = {a: a âˆˆ S and a âˆ‰ A} = S - A

If events A and B are associated with sample space S, then event "A or B" contains all the elements that are either in A or B, or in both.

It is denoted by A È B.

Event "A or B" = A È B = {a: a âˆˆ A or a âˆˆ B}

Consider Sample space S = {1, 2, 3, 4, 5, 6}

A = {1, 2, 5}

B = {2, 4, 6}

A È B = {1, 2, 4, 5, 6}

Two events A and B in a sample space S are said to be exhaustive if A âˆª B = S.

Events E1, E2, E3,..., En in a sample space S are said to be exhaustive if

E1âˆª E2 âˆª E3 âˆª...âˆª En = â‹ƒ i=1 n E i = S

Events E1, E2, ..., En are said to be exhaustive, if at least one of them necessarily occurs whenever the experiment is performed.

If events A and B are associated with sample space S, then event "A and B" contains the elements that are common to both A and B.

It is denoted by A âˆ© B.

Event "A and B" = A âˆ© B = {a: a âˆˆ A and  a âˆˆ B}

Consider a sample space of rolling two dice.

S = {(x,y): x,y = 1, 2, 3, 4, 5, 6}

Let events A and B be associated with S.

A: Event that the sum of the numbers on both the dice is at least 10.
B: Event that the number on the second die is 6.

The outcomes of events A and B:

A = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}

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

A Ç B = {(4, 6), (5, 6), (6, 6)}

Two events A and B in a sample space S are said to be mutually exclusive if A âˆ© B = f or A and B are disjoint sets.

Events E1, E2, E3,..., En in a sample space S are said to be exhaustive if

Some mutually exclusive events in a sample space are exhaustive.

Ex: In rolling a die,

S = {1, 2, 3, 4, 5, 6}

Event "getting even numbers" = {2, 4, 6}

Event "getting odd numbers" = {1, 3, 5}

{2, 4, 6} âˆ© {1, 3, 5} = âˆ…

{2, 4, 6} âˆª {1, 3, 5} = {1, 2, 3, 4, 5, 6} = S

Such events are called mutually exclusive and exhaustive.

In a sample space S, if Ei âˆ© Ei = âˆ… for i â‰  j and â‹ƒ i=1 n E i = S , then E1, E2, E3,..., En are called mutually exclusive and exhaustive events.

If events A and B are associated with sample space S, then event 'A but not B' is the event of all the elements of A but not in B. It is denoted by A - B.

Event "A but not B" = A - B = {a: a âˆˆ A and  a âˆˆ B}.

A - B = A âˆ© B'

Ex: Consider an experiment of throwing two dice simultaneously.

Consider events A and B associated with the experiment.

A: Sum of the numbers on both the dice is at least 9.

B: Sum of the numbers on both the dice is 10.

The outcomes of events A and B are:

A = {(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)}

B = {(4, 6), (5, 5), (6, 4)}

Event "A but not B" = A - B = {(3, 6), (4, 5), (5, 4), (5, 6), (6, 3), (6, 5), (6, 6)}.