Probability : Mutually-Exclusive/Incompatible & Non-Mutually-Exclusive/Compatible Events :: Probability

Two or more events are said to be mutually exclusive or incompatible when only one of those events can occur at a time.

No two of these events occur simultaneously i.e. the occurrence of one prevents the occurrence of the others.

The "Mutually Exclusivity" property gives an idea of the inter relationships between the events i.e. whether they are connected or not. Events which are mutually exclusive are not connected.

Example

1. In the experiment of tossing a coin:
   • The two possible elementary events are mutually exclusive. Where
     1. A : the event of getting a HEAD
     2. B : the event of getting a TAIL

     We say events A and B are mutually exclusive.

2. In the experiment of throwing a dice:
   • The six possible elementary events of getting 1, 2, 3, 4, 5 or 6 are mutually exclusive. Where
     1. A : the event of getting 1
     2. B : the event of getting 2
     3. .. .
     4. ...
     5. F : the event of getting 6

     We say events A, B, C, D, E and F are mutually exclusive.

   • The two compound events of getting an odd number and getting an even number are mutually exclusive.
     1. M : the event of getting an odd number
     2. K : the event of getting an even number

     We say events M and K are mutually exclusive.

For two or more events to be mutually exclusive, the only requirement is that they should not occur together. If one occurs, the other(s) should not occur.

Any two or more events satisfying this condition in relation to an experiment can be called mutually exclusive events.

It is not a requirement that these events should represent all possible events in the experiment.

Only some events of all the possible events can be said to be mutually exclusive, if they satisfy this condition.

Example

1. In the experiment of tossing/throwing a die:
   • The three elementary events of getting 1, 4, 5 are mutually exclusive. Where
     1. P : the event of getting 1
     2. Q : the event of getting 4
     3. R : the event of getting 5

     We say events P, Q and R are mutually exclusive.

     The events of getting 2, 3 and 6 are not considered here.

   • The following three compound events are mutually exclusive
     1. E : the event of getting a number less than 4 { 1, 2, 3}
     2. F : the event of getting an odd number {1, 3, 5}
     3. G : the event of getting 6

     We say events E, F and G are mutually exclusive.

     The events of getting 1 and getting 4 are not considered here.

Two or more events are said to be "Not Mutually Exclusive" or "Compatible" if they are not "Mutually Exclusive".

Two or more of these events can occur simultaneously. i.e. the occurrence of one does not prevent the occurrence of the others in all cases.

Example

1. In the experiment of throwing a die:

   • Compatible Event

   The event of getting an even number and the event of getting a number greater than 3 are not mutually exclusive i.e. they are compatible.

   Both the events occur when we get either 4 or 6. Where

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

   We say events M and N are not Mutually Exclusive or M and N are Compatible Events.

   • Non - Compatible (⇒ Mutually Exclusive) Event

   The five elementary events of getting 1, 2, 4, 5 or 6 are mutually exclusive. Where

   1. A : the event of getting 1
   2. B : the event of getting 2
   3. C : the event of getting 4
   4. D : the event of getting 5
   5. F : the event of getting 6

   We say events A, B, C, D, F are mutually exclusive.