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

## Mutually Exclusive or Incompatible Events

### Mutually Exclusive

#### Meaning

1. Unable to be both true at the same time.

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.

### Examples

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 as they cannot occur together. Occurrence of one of these events implies non-occurrene of the other.

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 as they cannot occur together. Occurrence of any one of these events implies non-occurrene of all the other events.

• The two compound events of getting an odd number and getting an even number are mutually exclusive.

Where

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 as they cannot occur together. Occurrence of one of these events implies non-occurrene of the other.

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.

## To be mutually exclusive - all possible events need not be taken into consideration

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 together 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.

### Examples

1. In the experiment of tossing/throwing a die:

There are six possible elementary events.

The events of getting 1, 2, 3, 4, 5, 6 on the face of the dice.

• 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 as they cannot occur together. Occurrence of one of these events implies non-occurrene of the other.

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

• The following three compound events are mutually exclusive

Where

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 as they cannot occur together. Occurrence of one of these events implies non-occurrene of the other.

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

## Non/Not Mutually Exclusive or Compatible Events

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.

### Examples

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/Incompatible/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.

Any two more of these events taken together would form mutually exclusive events.

## Pairwise Exclusive Events

Three or more events are said to be Pairwise mutually exclusive if no two of them are compatible i.e. every two events considered separately would be mutually exclusive.

Three or more events in consideration, would be mutually exclusive if and only if they are pairwise mutually exclusive.

Mutually Exclusive ⇔ Pair wise Exclusive.

### Examples

1. In the experiment of tossing/throwing a die:

If

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

The events A, B, C considered pairwise would give the following pairs of events A & B, B & C and C & A.

1. Events 'A' and 'B' are mutually exclusive,
2. Events 'B' and 'C' are mutually exclusive,
3. Events 'C' and 'A' are mutually exclusive,
Since events A, B and C are pairwise mutually exclusive, A, B and C together can be said to be mutually exclusive.

2. In the experiment of tossing/throwing a die:

If

1. P : the event of getting an even number
2. Q : the event of getting an odd number
3. R : the event of getting a number greater than 5

The events P, Q, R considered pairwise would give the following pairs of events P & Q, Q & R and R & P.

1. Events 'P' and 'Q' are mutually exclusive,
2. Events 'Q' and 'R' are mutually exclusive,
3. Events 'R' and 'P' are not mutually exclusive,
[Both the events occur when 6 appears on the die.]
Since events P, Q and R are not pairwise mutually exclusive, P, Q and R together cannot be said to be mutually exclusive.

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