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

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 or compound events.

Example

1. In the experiment of tossing a coin:

   Equally Likely

   Where
   1. A : the event of getting a "HEAD" and
   2. 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].

2. In the experiment of throwing a die:

   Equally Likely

   Where
   1. A : the event of getting 1
   2. B : the event of getting 2
   3. ...
   4. ...
   5. 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
   1. M : the event of getting an even number
   2. 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
   1. P : the event of getting an odd number {1, 3, 5}
   2. 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.

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.

Example

1. In the experiment of throwing a die:

   Equally Likely & Not Mutually Exclusive

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

   The two events "G" and "H"

   • are equally likely.
     [as each would occur on the occurrence of three possible elementary events]
   • are not mutually exclusive.
     [both event "G" and "H" occur when "2" appears on the die]

   Equally Likely & Mutually Exclusive

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

   The two compound events "M" and "N"

   • are equally likely.
     [as each would occur on the occurrence of three possible elementary events]
   • are mutually exclusive.
     [as they do not occur together]

   Not Equally Likely & Not Mutually Exclusive

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

   The two compound events "P" and "Q"

   • are not equally likely.
     [as the first event would occur on the occurrence of two possible elementary events
     and the second on the occurrence of three possible elementary events]
   • are not mutually exclusive.
     [as both the events would occur when 1 or 5 appear on the dice]

   Not Equally Likely & Mutually Exclusive

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

   The two compound events "P" and "Q"

   • are not equally likely.
     [as the first event would occur on the occurrence of three possible elementary events
     and the second on the occurrence of two possible elementary events]
   • are mutually exclusive.
     [as both the events would not occur together]

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.

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

Example

1. In the experiment of tossing a coin:

   Exhaustive

   Where

   1. A : the event of getting a HEAD
   2. B : the event of getting a TAIL

   The two events "A" and "B" are called exhaustive events.
   [When we conduct the experiment, at least one of these will occur.]

2. In the experiment of throwing a die:

   Exhaustive

   Where

   1. A : the event of getting 1
   2. B : the event of getting 2
   3. ...
   4. ...
   5. 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

   1. L : the event of getting an even number
   2. 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.]

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. 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.

Example

1. In the experiment of tossing a coin:

   Not Exhaustive

   Where
   1. A : the event of getting a HEAD
   2. 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.
2. In the experiment of throwing a die:

   Not Exhaustive

   Where
   1. A : the event of getting 1
   2. B : the event of getting 2
   3. ...
   4. ...
   5. 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.

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

Example

1. In the experiment of tossing a coin:

   Where

   1. 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.

2. In the experiment of rolling a die:

   Where

   1. 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

• One Event : Exhaustive ⇒ Certain Event

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

Example

1. In the experiment of rolling a dice:

   Where

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

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

   Any events combined with Exhaustive events will also form Exhaustive Events

   1. C : the event of getting a number greater than "2" {3, 4, 5, 6}

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

   2. 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.

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

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

Example

1. In the experiment of throwing a die:

   Exhaustive & Not Mutually Exclusive

   Where

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

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

   • are not mutually exclusive
     [as both "G" and "I" occur when "2" appears]
   • are exhaustive
     [they cover all the possible elementary events over them]

   Exhaustive & Mutually Exclusive

   Where

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

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

   • are mutually exclusive
     [as they do not occur together]
   • are exhaustive
     [as they cover all the possible elementary events over them]

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

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

Example

1. In the experiment of throwing a die:

   Exhaustive & Not Equally Likely

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

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

   • are mutually exhaustive
     [as they cover all the possible elementary events over them]
   • are not equally likely
     [as "G" and "I" are three times more likely to occur than "H". ]

   Exhaustive & Equally Likely

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

   The two compound events "M" and "N"

   • are mutually exhaustive
     [as they cover all the possible elementary events over them]
   • are equally likely
     [as they have an equally likely chance of occurrence]