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Events

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Events - Lesson Summary

Consider an experiment of throwing a die once.

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

The outcomes that correspond to the occurrence of even numbers: 2, 4, 6.
The outcomes that correspond to the occurrence of odd numbers: 1, 3, 5.
Such outcomes are called events.

An outcome or a collection of outcomes is called an event.

Let's the events be denote by E and O.
E = {2, 4, 6}, O = {1, 3, 5}

Sets E and O are the subsets of set S.
Any subset of a sample space is called an event.

If S is a sample space, then ∅ ⊆ S and S ⊆ S.

Hence, subsets ∅ and S are two events in S.

Consider an experiment of rolling a die.

Let E be an event "a number less than three appears".

If the outcome is either one or two, then we say that event E has occurred.

If the outcome is other than one or two, then we say that event E has not occurred.

Types of Events
Events can be classified into sure, impossible, simple and compound events based on the elements they have.

Sure event
Consider an experiment, where a coin is tossed. The occurrence of an event of a head or tail is sure. Such events are called sure events.

If S is a sample space, then event S is called a sure event.

Examples:
•  In dropping a ball from a height, the event that it will hit the ground.
•  In throwing a die, the event of getting a number from 1 to 6.

Impossible event
In an experiment, a die is thrown.
In this case, the event of getting a number that is divisible by seven is not possible.

Such events are called impossible events.

The event "null set of a sample space" will never happen.

If S is a sample space, then the event ∅ is called an impossible event.

Examples:

•   The event of selecting an apple from a bag consisting of 10 oranges.
•   When a ball is thrown up, the event that the ball will stay in the air.

Simple events
Consider a sample space S of an experiment of rolling a die and its associated events.
S = {1, 2, 3, 4, 5, 6}
E1 = {1} E2 = {2} E3 = {3}
E4 = {4} E5 = {5} E6 = {6}

An event of a sample space with only one sample point is called a simple or elementary event.

The number of simple events of a sample space of n sample points is n.

Compound event

An event that has more than one sample point is called a compound event.

Consider the sample space of an experiment of tossing two coins.

S = {HH, HT, TH, TT}

Events E1 and E2 associated with sample space S are E1 = {HH, TT} and E2 = {HH, HT, TH}.

Both the events have more than one sample point. Therefore, the events are compound events.

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