Two or more events which have an equally likely chance or 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.

Examples

1. In the experiment of tossing a coin:

   Where

   A : the event of getting a HEAD

   B : the event of getting a TAIL

   Equally Likely

   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:

   Where

   A : the event of getting 1

   B : the event of getting 2

   ...

   F : the event of getting 6

   Equally Likely

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

   All these events have the same chance of occurrence.

   Where

   M : the event of getting an odd number

   N : the event of getting an even number

   Equally Likely

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

   Where

   P : the event of getting an odd number {1, 3, 5}

   Q : the event of getting 6

   Not Equally Likely

   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.

Examples

1. In the experiment of throwing a die:

   Where

   G : the event of getting a prime number {2, 3, 5}

   H : the event of getting an even number {2, 4, 6}

   Equally Likely & Not Mutually Exclusive

   The two events G and H

   • are equally likely.

     each would occur on the occurrence of three possible elementary events

   • are not mutually exclusive.

     both the events occur when 2 appears on the die

   Where

   M : the event of getting an even number {2, 4, 6}

   N : the event of getting an odd number {1, 3, 5}

   Equally Likely & Mutually Exclusive

   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

   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}

   Not Equally Likely & Not Mutually Exclusive

   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

   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}

   Not Equally Likely & Mutually Exclusive

   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.

Examples

1. In the experiment of tossing a coin:

   Where

   A : the event of getting a HEAD

   B : the event of getting a TAIL

   Exhaustive

   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:

   Where

   A : the event of getting 1

   B : the event of getting 2

   ...

   F : the event of getting 6

   Exhaustive

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

   One of these events will occur whenever the experiment is conducted.

   Where

   L : the event of getting an even number

   M : the event of getting an odd number

   Exhaustive

   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.

Examples

1. In the experiment of tossing a coin:

   Where

   A : the event of getting a HEAD

   B : the event of getting a TAIL

   Not Exhaustive

   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:

   Where

   A : the event of getting 1

   B : the event of getting 2

   ...

   F : the event of getting 6

   Not Exhaustive

   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.

Only a single event can form an exhaustive event.

Examples

1. In the experiment of tossing a coin:

   Where

   A : the event of getting either a HEAD or a TAIL

   Exhaustive

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

2. In the experiment of rolling a die:

   Where

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

   Exhaustive

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

It is therefore a Certain Event.

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

Examples

1. 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.

   Any events combined with Exhaustive events will also 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.

   • Where

     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.

Examples

1. In the experiment of throwing a die:

   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}

   Exhaustive & Not Mutually Exclusive

   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

   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

     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.

Examples

1. In the experiment of throwing a die:

   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}

   Exhaustive & Not Equally Likely

   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.

   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}

   Exhaustive & Equally Likely

   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