Events and Types of Events in Probability: Impossible and Sure Events

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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.

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 and .

Events in Probability

Events in Probability

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

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:

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 favorable 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:

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 and 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 then, and 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 and are two events such that consists of numbers less than 3 and consists of numbers greater than 4.

So,

Then, and 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 , which represents the remaining elements of the sample space S.

If a dice is rolled then the sample space S is given as If event E1 represents all the outcomes which is greater than 4, then and

Thus, is the complement of the event

Similarly, the complement of will be represented as

Events Associated with “Or”

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

Thus, the event denotes

If we have mutually exhaustive events associated with sample space S then,

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 denotes

Event but Not

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

difference between both the events

Difference between Both the Events

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Example Question on Probability of Events

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

i)

ii)

iii)

Solution:

The sample space is given as

i)

ii)

iii)

Question:

When a fair dice is thrown, what is the probability of getting

(a) the number 5

(b) a number that is a multiple of 3

(c) a number that is greater than 6

(d) a number that is less than 7

Solution:

A fair die is an unbiased die where each of the six numbers is equally likely to turn up.

(a) Let event of getting the number 5

Let number of outcomes in event

number of outcomes in

(b) Let event of getting a multiple of 3

Multiple of

(c) Let event of getting a number greater than 6

There is no number greater than 6 in the sample space S.

A probability of 0 means the event will never occur.

(d) Let event of getting a number less than 7

Number less than

A probability of 1 means the event will always occur.

FAQs

What Are Events in Probability?

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.

What is the Difference between Sample Space and Event?

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 and the event of getting an even number will be

What is the Probability of an Impossible Event and a Sure Event?

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

What is an Example of an Impossible Event?

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