## Equally Likely, Exhaustive, Not-Exhaustive Events :: Probability |

## Equally Likely, Exhaustive, Not-Exhaustive Events :: Probability |

## Equally likely events |

Two or more events which have an equal probability of occurrence are said to be equally likely, i.e. if on taking into account all the conditions, there should be no reason to except any one of the events in preference over the others.

Equally likely events may be elementary events or compound events.

- In the experiment of tossing a coin:
#### Equally Likely

Where- A : the event of getting a "HEAD" and
- B : the event of getting a "TAIL"

Events "A" and "B" are said to be equally likely events

[Both the events have the same chance of occurrence]. - In the experiment of throwing a die:
#### Equally Likely

Where- A : the event of getting 1
- B : the event of getting 2
- ...
- ...
- F : the event of getting 6

Events "A", "B", "C", "D", "E", "F" are said to be equally likely events

[All these events have the same chance of occurrence.]#### Equally Likely

Where- M : the event of getting an even number
- N : the event of getting an odd number

The two compound events "M" and "N" are said to be equally likely.

#### Not Equally Likely

Where- P : the event of getting an odd number {1, 3, 5}
- Q : the event of getting 6

The two events "P" and "Q" cannot be said to be equally likely.

Event "P" occurs when any of the elementary events of getting "1", "3" and "5" occur.

Event "Q" occurs only when the elementary event of getting "6" occur.

Event "P" is three times more likely to occur than "Q".

⇒ "P" and "Q" are not equally likely.

## Equally likely & Mutually Exclusive are distinct characteristics |

The event characteristics equally likely and mutually exclusive are independent of each other.

Equally likely events may be mutually exclusive or not mutually exclusive.

⇒ Mutually Exclusive events may be equally likely or not equally likely.

- In the experiment of throwing a die:
#### Equally Likely & Not Mutually Exclusive

Where- G : the event of getting a prime number {2, 3, 5}
- H : the event of getting an even number {2, 4, 6}

The two events "G" and "H" are

##### Equally likely

Each would occur on the occurrence of three possible elementary events.##### Not mutually exclusive

Both event "G" and "H" occur when "2" appears on the die.

#### Equally Likely & Mutually Exclusive

Where- M : the event of getting an even number {2, 4, 6}
- N : the event of getting an odd number {1, 3, 5}

The two compound events "M" and "N" are

##### Equally likely

Each would occur on the occurrence of three possible elementary events.##### Mutually exclusive

They do not occur together.

#### Not Equally Likely & Not Mutually Exclusive

Where- P : the event of getting one of the numbers 1, 5 {1, 5}
- Q : the event of getting an odd number {1, 3, 5}

The two compound events "P" and "Q", are

##### Not Equally likely

The first event would occur on the occurrence of two possible elementary events and the second on the occurrence of three possible elementary events.##### Not Mutually Exclusive

Both the events would occur when 1 or 5 appear on the dice.

#### Not Equally Likely & Mutually Exclusive

Where- K : the event of getting an even number {2, 4, 6}
- L : the event of getting one of the numbers 3, 5 {3, 5}

The two compound events "P" and "Q" are

##### Not Equally Likely

Event "P" would occur on the occurrence of three possible elementary events and event "Q" would occur on the occurrence of two possible elementary events.##### Mutually Exclusive

Both the events would not occur together.

## Exhaustive Events |

One or more events are said to be exhaustive if all the possible elementary events under the experiment are covered by the event(s) considered together. In other words, the events are said to be exhaustive when they are such that at least one of the events compulsorily occurs. ### Examples

Exhaustive events may be elementary or compound events. They may be equally likely or not equally likely.

- In the experiment of tossing a coin:
#### Exhaustive

Where- A : the event of getting a HEAD
- B : the event of getting a TAIL

The two events "A" and "B" together are called exhaustive events.

[When we conduct the experiment, at least one of these will occur.] - In the experiment of throwing a die:
#### Exhaustive

Where- A : the event of getting 1
- B : the event of getting 2
- ...
- ...
- F : the event of getting 6

The six Events "A", "B", "C", "D", "E", "F" together are called exhaustive events.

[One of these events will occur whenever the experiment is conducted.]#### Exhaustive

Where- L : the event of getting an even number
- M : the event of getting an odd number

The two compound events "L" and "M" together are said to be exhaustive events.

[One of the events will occur whenever the experiment is conducted.]

## Non/Not Exhaustive Events |

One or more events are said to be not exhaustive if all the possible elementary events under the experiment are not covered by the event(s) considered together. ### Examples

In other words, the events are said to be not exhaustive when they are such that there is at least one elementary event in the experiment that does not form a part of these events taken together.

Where the events taken together do not form exhaustive events they are "Not Exhaustive" events.

- In the experiment of tossing a coin:
#### Not Exhaustive

Where- A : the event of getting a HEAD
- B : the event of getting a TAIL

If we consider only Event "A", it is Not Exhaustive.

[It does not cover all the possible choices. The event of getting a TAIL is not covered]If we consider only Event "B", it is Not Exhaustive.

[It does not cover all the possible choices. The event of getting a HEAD is not covered]#### Exhaustive

Events "A" and "B" together would form Exhaustive events. - In the experiment of throwing a die:
#### Not Exhaustive

Where- A : the event of getting 1
- B : the event of getting 2
- ...
- ...
- F : the event of getting 6

Events "A", "B", "C", "D", "F" together would not form exhaustive events.

Events "B", "D", "E" together would not form exhaustive events

Any five or less of these events together do not form exhaustive events since they do not cover all the possible outcomes. Whenever the experiment is conducted we cannot for sure say that one of the five events would occur as the sixth may also occur.

#### Exhaustive

All the possible events ("A", "B", ..., "F") considered together would form exhaustive events.

## If a Single Event is exhaustive, it is a Certain Event |

A single event forms an exhaustive event if it happens for sure whenever the experiment is conducted. ### Examples

### If an event is exhaustive it is a Certain Event

- In the experiment of tossing a coin:
#### Exhaustive

Where- A : the event of getting either a "HEAD" or a "TAIL"

We say "A" is an exhaustive event as it occurs whenever the experiment is conducted.

- In the experiment of rolling a die:
#### Exhaustive

Where- M : the event of getting any number between 0 and 7

"M" is an exhaustive event as it occurs (we get a number between 0 and 7) whenever the experiment is conducted.

Where one event forms an exhaustive event, it is bound to happen for sure.

It is therefore a "Certain Event"

## Event(s) taken together with exhaustive events will form exhaustive events |

Where one or more events are already exhaustive, any other events (one or more other) combined together with these would always be exhaustive. ### Examples

- In the experiment of rolling a dice:
Where

- A : the event of getting an even number {2, 4, 6}
- B : the event of getting an odd number {1, 3, 5}

"A" and "B" together will form exhaustive events.

#### One or more events combined with Exhaustive events will form Exhaustive Events

Where- C : the event of getting a number greater than "2" {3, 4, 5, 6}
"A", "B" and "C" together would form exhaustive events.

- M : the event of getting a prime number {2, 3, 5}
"A", "B" and "M" together would form exhaustive events.

"A", "B" "C" and "M" together would form exhaustive events.

## Exhaustive events may be either mutually exclusive or not mutually exclusive |

The event characteristics "Exhaustive" events and "Mutually Exclusive" Events are independent of each other. ### Examples

"Exhaustive Events" may be "Mutually Exclusive" or "Not Mutually Exclusive".

- In the experiment of throwing a die:
#### Exhaustive & Not Mutually Exclusive

Where- G : the event of getting a prime number {2, 3, 5}
- H : the event of getting 1 {1}
- I : the event of getting an even number {2, 4, 6}

The three events "G", "H" and "I" are

##### Not Mutually Exclusive

Both "G" and "I" occur when "2" appears.##### Exhaustive

They cover all the possible elementary events over them.

#### Exhaustive & Mutually Exclusive

Where- M : the event of getting an even number {2, 4, 6}
- N : the event of getting an odd number {1, 3, 5}

The two compound events "M", and "N" are

##### Mutually Exclusive

They do not occur together.##### Exhaustive

They cover all the possible elementary events over them.

## Exhaustive events may be either equally likely or not equally likely |

The event characteristics "Exhaustive Events" and "Not Equally Likely" events are independent of each other. ### Examples

"Exhaustive" events may be "Equally Likely" events or "Not Equally Likely" events.

- In the experiment of throwing a die:
#### Exhaustive & Not Equally Likely

Where- G : the event of getting a prime number {2, 3, 5}
- H : the event of getting 1 {1}
- I : the event of getting an even number {2, 4, 6}

The three events "G", "H" and "I" are

##### Exhaustive

They cover all the possible elementary events over them.##### Not Equally Likely

"G" and "I" are three times more likely to occur than "H".

#### Exhaustive & Equally Likely

Where- M : the event of getting an even number {2, 4, 6}
- N : the event of getting an odd number {1, 3, 5}
- I : the event of getting an even number {2, 4, 6}

The two compound events "M" and "N" are

##### Exhaustive

They cover all the possible elementary events over them.##### Equally Likely

They have an equally likely chance of occurrence.

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