- Category: Mathematics

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.

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.

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**

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

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.

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.

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}, E_{1} = {56 ,54 }, E_{2} = {78 ,56 ,89 } then, E_{1} and E_{2} represent two compound 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**.

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 E_{1}, E_{2} are two events such that E_{1} consists of numbers less than 3 and E_{2} consists of numbers greater than 4.

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

Then, E1 and E2 are mutually exclusive.

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

For any event E_{1} there exists another event E_{1}‘ which represents the remaining elements of the sample space S.

**E _{1} = S − E_{1}‘**

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

Thus E_{1}‘ is the complement of the event E_{1}.

Similarly, the complement of E_{1}, E_{2}, E_{3}……….E_{n }will be represented as E_{1}‘, E_{2}‘, E_{3}‘……….E_{n}‘

If two events E_{1} and E_{2} are associated with **OR** then it means that either E_{1} or E_{2} or both. The union symbol **(∪)** is used to represent OR in probability.

Thus, the event E_{1}, E_{2} denotes E_{1}, E_{2}.

If we have mutually exhaustive events E_{1}, E_{2}, E_{3 }………E_{n} associated with sample space S then,

E_{1}, E_{2}, E_{3}………E_{n} = S

If two events E_{1} and E_{2} 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 E_{1}, E_{2} denotes E_{1} and E_{2}.

It represents the difference between both the events. Event E_{1 }but not E_{2} represents all the outcomes which are present in E_{1} but not in E_{2}. Thus, the event E_{1} but not E_{2} is represented as

E_{1}, E_{2} = E_{1} E_{2}

**Question:** In the game of snakes and ladders, a fair die is thrown. If event E_{1} represents all the events of getting a natural number less than 4, event E_{2} consists of all the events of getting an even number and E_{3} denotes all the events of getting an odd number. List the sets representing the following:

i) E_{1} or E_{2} or E_{3}

ii) E_{1} and E_{2} and E_{3}

iii) E_{1} but not E_{3}

**Solution:**

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

E_{1} = {1,2,3}

E_{2} = {2,4,6}

E_{3} = {1,3,5}

i)E_{1} or E_{2} or E_{3}= E_{1} E_{2} E_{3}= {1, 2, 3, 4, 5, 6}

ii)E_{1} and E_{2} and E_{3} = E_{1} E_{2} E_{3} = ∅

iii)E_{1} but E_{3} = E_{1} E_{2}= {2}

In probability, events are the outcomes of an experiment. The probability of an event is the measure of the chance that the event will occur as a result of an experiment.

A sample space is a collection or a set of possible outcomes of a random experiment while an event is the subset of sample space. For example, if a die is rolled, the sample space will be {1, 2, 3, 4, 5, 6} and the event of getting an even number will be {2, 4, 6}.

The probability of a sure event is always 1 while the probability of an impossible event is always 0.

An example of an impossible event will be getting a number greater than 6 when a die is rolled.