Events and Types of Events in Probability

What are Events in Probability?

A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space. So, what is sample space?

The entire possible set of outcomes of a random experiment is the sample space or the individual space of that experiment. The likelihood of occurrence of an event is known as probability. The probability of occurrence of any event lies between 0 and 1.

Events In Probability

The sample space for the tossing of three coins simultaneously is given by:

S = {(T , T , T) , (T , T , H) , (T , H , T) , (T , H , H ) , (H , T , T ) , (H , T , H) , (H , H, T) ,(H , H , H)}

Suppose, if we want to find only the outcomes which have at least two heads; then the set of all such possibilities can be given as:

E = { (H , T , H) , (H , H ,T) , (H , H ,H) , (T , H , H)}

Thus, an event is a subset of the sample space, i.e., E is a subset of S.

There could be a lot of events associated with a given sample space. For any event to occur, the outcome of the experiment must be an element of the set of event E.

What is the Probability of Occurrence of an Event?

The number of favourable outcomes to the total number of outcomes is defined as the probability of occurrence of any event. So, the probability that an event will occur is given as:

P(E) = Number of Favourable Outcomes/ Total Number of Outcomes

Types of Events in Probability:

Some of the important probability events are:

• Impossible and Sure Events
• Simple Events
• Compound Events
• Independent and Dependent Events
• Mutually Exclusive Events
• Exhaustive Events
• Complementary Events
• Events Associated with “OR”
• Events Associated with “AND”
• Event E1 but not E2

Impossible and Sure Events

If the probability of occurrence of an event is 0, such an event is called an impossible event and if the probability of occurrence of an event is 1, it is called a sure event. In other words, the empty set ϕ is an impossible event and the sample space S is a sure event.

Simple Events

Any event consisting of a single point of the sample space is known as a simple event in probability. For example, if S = {56 , 78 , 96 , 54 , 89} and E = {78} then E is a simple event.

Compound Events

Contrary to the simple event, if any event consists of more than one single point of the sample space then such an event is called a compound event. Considering the same example again, if S = {56 ,78 ,96 ,54 ,89}, E₁ = {56 ,54 }, E₂ = {78 ,56 ,89 } then, E₁ and E₂ represent two compound events.

Independent Events and Dependent Events

If the occurrence of any event is completely unaffected by the occurrence of any other event, such events are known as an independent event in probability and the events which are affected by other events are known as dependent events.

Mutually Exclusive Events

If the occurrence of one event excludes the occurrence of another event, such events are mutually exclusive events i.e. two events don’t have any common point. For example, if S = {1 , 2 , 3 , 4 , 5 , 6} and E₁, E₂ are two events such that E₁ consists of numbers less than 3 and E₂ consists of numbers greater than 4.

So, E1 = {1,2} and E2 = {5,6} .

Then, E1 and E2 are mutually exclusive.

Exhaustive Events

A set of events is called exhaustive if all the events together consume the entire sample space.

Complementary Events

For any event E₁ there exists another event E₁‘ which represents the remaining elements of the sample space S.

E₁ = S − E₁‘

If a dice is rolled then the sample space S is given as S = {1 , 2 , 3 , 4 , 5 , 6 }. If event E₁ represents all the outcomes which is greater than 4, then E₁ = {5, 6} and E₁‘ = {1, 2, 3, 4}.

Thus E₁‘ is the complement of the event E₁.

Similarly, the complement of E₁, E₂, E₃……….E_n will be represented as E₁‘, E₂‘, E₃‘……….E_n‘

Events Associated with “OR”

If two events E₁ and E₂ are associated with OR then it means that either E₁ or E₂ or both. The union symbol (∪) is used to represent OR in probability.

Thus, the event E₁U E₂ denotes E₁ OR E₂.

If we have mutually exhaustive events E₁, E₂, E₃ ………E_n associated with sample space S then,

E₁ U E₂ U E₃U ………E_n = S

Events Associated with “AND”

If two events E₁ and E₂ are associated with AND then it means the intersection of elements which is common to both the events. The intersection symbol (∩) is used to represent AND in probability.

Thus, the event E₁ ∩ E₂ denotes E₁ and E₂.

Types of Events In Probability

Event E₁ but not E₂

It represents the difference between both the events. Event E₁ but not E₂ represents all the outcomes which are present in E₁ but not in E₂. Thus, the event E₁ but not E₂ is represented as

E₁, E₂ = E₁ – E₂

Example Question on Probability of Events

Question: In the game of snakes and ladders, a fair die is thrown. If event E₁ represents all the events of getting a natural number less than 4, event E₂ consists of all the events of getting an even number and E₃ denotes all the events of getting an odd number. List the sets representing the following:

i)E₁ or E₂ or E₃

ii)E₁ and E₂ and E₃

iii)E₁ but not E₃

Solution:

The sample space is given as S = {1 , 2 , 3 , 4 , 5 , 6}

E₁ = {1,2,3}

E₂ = {2,4,6}

E₃ = {1,3,5}

i)E₁ or E₂ or E₃= E₁ E₂ E₃= {1, 2, 3, 4, 5, 6}

ii)E₁ and E₂ and E₃ = E₁ E₂ E₃ = ∅

iii)E₁ but not E₃ = {2}

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