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Union and Intersections

Statistics Assignment / Homework Help
Union and Intersection

We have discussed about mutually exclusive and independent events. Now just think about this question: "Can we conclude that all independent events aremutually exclusive?"

Let us try to answer this question using simple examples.

Consider the random experiment of birth of a child.

Then our sample space is {boy, girl}

Let event A: birth of a son

A={son}

Let event B: birth of a daughter

Then B={daughter}

Now we know that the events A and B are independent, since the birth of a son or daughter does not depend on one other. But they are not mutually exclusive, because in one pregnancy both a daughter and son can be born. Hence we can conclude that all independent events are not mutually exclusive ones.

Now what about the inverse of this- "Can we conclude that all mutually exclusive event s are independent?"

From the above example it implies that the converse is also not true.

Equally likely events: two events A and B are said to be equally likely if both have the same chance of occurrence.

For example consider the random experiment of tossing a coin

Then our sample space is S={H,T}

Consider event A: getting a head

Then A={H}

Consider event B: getting a tail

Then B={T}

Now both events A and B have equal chance of occurrence. This is because since the coin is a fair one, either head or tail can appear. We cannot predict the outcome of the experiment. Hence A and B are equally likely events.

Union of events: consider two events A and B. the union of A and B is the occurrence of atleast either A or B. to put it in simple terms, the occurrence of atleast one of the events is called union of events.

Union of Any two events A and B is the occurrence of either A or B.

The union of events A and B is represented by AUB. In mathematical terms union of two events A and B is represented B is represented by A+B. we can also term union of A and B as the occurrence of A or B.

Consider the random experiment of tossing a die.

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

Let event A: getting a multiple of 2

A={2,4,6}

Let event B: getting a multiple of 5

B={5}

Now consider the event AUB. It is given by

AUB={2,4,5,6}

When we see the event AUB, we can understand that either event A or event B has occurred.

Intersection of events: two or more events are said to be intersecting if all of them occur simultaneously.

The intersection of events is the event of occurrence of all of them. If we consider two events A and B, then the simultaneous occurrence if both A and B is called the intersection of A and B. the intersection of two events A and B is denoted as AᴖB. in other words the occurrence of both a and B is called the intersection of A and B. in mathematical terms the intersection of two events A and B is termed as A*B.

Consider the random experiment of selecting a number form the first ten natural numbers.

Then our sample space is S={1,2,3,4,5,6,7,8,9,10}

Let event A: selecting a multiple of 4

Then A={4,8}

Let event B:selecting and multiple of 2

Then B=[2,4,6,8,10}

Now the event AᴖB={4,8}-that is only the common elements in A and B.

Now we can define mutually exclusive events even by using the concept of intersection of events.

"Two events A and B are said to be mutually exclusive if AᴖB is a null event"

Null event is the event which does not have any elements in it.

Do both the definitions of mutually exclusive mean the same? Think about it. Take a simple example and try to understand if both the definitions mean the same.

When we learn about probability and its calculations, we will be using the above concepts extensively. Probability places a major role in medical and economical field. Though the concept is quite complicated, it has a large number of implications. Because of this there has been a innumerable developments in this field. Many eminent statisticians like James Bernoulli, De Moivre, Laplace, etc have contributed a lot to this theory. James Bernoulli is considered the pioneer of this field.

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