Union and Intersection of Events

A union refers to an area that is said to be belonging to one or both of two events. In contrast, an intersection is an area that is said to be belonging to both of those two events. You can write down the union of two events as A∪B. On the other hand, you can write down the intersection of two events as A∩B. In order to perform basic probability calculations, understanding the union and intersection of events is essential.

Union of Events

The union of two or more sets refers to the set with all the elements belonging to each set. An element is said to be in the union if it lies to at least one of the sets. The symbol ∪ is popularly used for union and its association with the word “or”. This is because A∪B is the set of all elements in A or B or even both.

In order to find out the union of two sets, we list the elements in A or B or both sets. This can be easily represented with the help of the Venn diagram. Here, the expression of the union of sets A and B can take place as two interlocking circles that are entirely shaded.

For a good understanding of union and intersection of events, one must understand their expression in symbols. In symbols, the definition of the union of events can be expressed as follows:

A∪B = {x : x ∈ A or x ∈ B}

So, if we consider, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6}

Then, A∪B would be = {1, 2, 3, 4, 5, 6, 7}

Here, the listing of element 1 does not take place twice in the union. Even though its appearance is in both A and B sets.

Intersection of Events

The intersection of two or more sets refers to the set of elements universal to each set. An element is in the intersection if it occurs in all of the sets. The symbol ∩ is used for intersection and its association with the word “and”. This is because A∩B is the set of elements simultaneously pertaining to A and B.

In order to determine the intersection of two or more than two sets, only those elements must be involved whose listing takes place in both or all of the sets. This can be easily shown with the help of the Venn diagram. Here, the representation of the intersection of two sets, A and B, can be the shaded region. This region can be in the middle of two interlocking circles.

In mathematical notation, the expression of the intersection of A and B can take place as follows:

As A∩B = {x : x ∈ A and x ∈ B}

So, if we consider A = {1, 3, 5, 7} and B = {1, 2, 4, 6}

Then A∩B = {1}

This is because 1 is the only element in both A and B sets.

Probability of the Union of Events

To compute this probability, the first thing to do is check their compatibility or incompatibility.

The probability of the union of incompatible events can be expressed as follows:

P(A∪B) = P(A) + P(B)

The probability of the union of compatible events can be expressed as follows:

P(A∪B) = P(A) + P(B) − P(A∩B)

In case of incompatible events, P(A∩B) = 0, the truth lies in the second formula.

Probability of the Intersection of Events

To calculate the probability of the intersection of events, we have to verify their dependence or independence.

The probability of the intersection of independent events can be expressed as follows:

P(A∩B) = P(A)⋅P(B)

The probability of the intersection of dependent events can be expressed as follows:

P(A∩B) = P(A/B)⋅P(B)

If the events are independent, P(A/B) = P(A), the truth lies in the second formula.

Conclusion

Union and intersection of events are two fundamental concepts of set theory. A union refers to an area belonging to one or both of two events. Here, all the elements belong to each set.  In contrast, an intersection is an area belonging to both events. It means a set of elements universal to each set. Union and intersection of events can be easily demonstrated by using a Venn diagram. You can compute the probability of the union of events by checking their compatibility or incompatibility. In case of the probability of the intersection of events, verify their dependence or independence.