The total number of possible outcomes of an experiment i.e. the number of elementary events possible in an experiment form the number of possible choices or cases in the experiment.

In its most common notation, this is represented by the letter n.

Examples

1. In the experiment of tossing a coin:

   There are two possible results or outcomes or choices i.e. there are two possible elementary events.

   1. The event of getting a HEAD
   2. The event of getting a TAIL

   Therefore the total number of possible choices in this experiment is 2

   ⇒ n = 2

2. In the experiment of tossing throwing a die:

   There are six possible results or outcomes or choices i.e. there are two possible elementary events.

   1. The event of 1 appearing on the face of the die
   2. The event of 2 appearing on the face of the die
   3. ...
   4. The event of 6 appearing on the face of the die

   Therefore the total number of possible choices in this experiment is 6

   ⇒ n = 6

The cases or choices or elementary events which ensure the occurrence of an event are called favorable cases or choices for the event.

Successes for an Event

The number of Favorable Choices for an event is also identified as the number of successes for the occurrence of the event.

In its most common notation, the number of Favorable choices or number of successes is represented by the letter m. Where we deal with more than one event at the same time, it is followed by a subscript (m_A, m_B etc) to indicate the event for which the choices are favorable.

Examples

1. In the experiment of tossing a coin:

   There are two possible results or outcomes or choices i.e. there are two possible elementary events.

   1. The event of getting a HEAD
   2. The event of getting a TAIL

   Where

   A : the event of getting a TAIL

   For Event A

   Number of favorable/favourable Choices (Or) Successes

   = 1 {TAIL}

   All elementary events of the experiment where we get a TAIL are favorable to the occurrence of Event A

   ⇒ m_A = 1

2. In the experiment of throwing a die:

   There are six possible outcomes or elementary events.

   1. The event of 1 appearing on the face of the die
   2. The event of 2 appearing on the face of the die
   3. ...
   4. ...
   5. ...
   6. The event of 6 appearing on the face of the die

   Where

   A : the event of getting an even number

   For Event A

   Number of favorable/favourable Choices (Or) Successes

   = 3 {TWO, FOUR, SIX}

   All elementary events of the experiment where we get an even number are favorable to the occurrence of Event A

   ⇒ m_A = 3

   Where

   P : the event of getting a number greater than 4

   For Event P

   Number of favorable/favourable Choices (Or) Successes

   = 2 {FIVE, SIX}

   All elementary events of the experiment where we get get a number greater than 4 are favorable to the occurrence of Event P

   ⇒ m_P = 2

The cases or choices or elementary events which ensure the non-occurrence of an event are called unfavourable/unfavorable cases or choices for the event.

Failures for an Event

The number of Unfavorable Choices for an event is also identified as the number of failures for the occurrence of the event.

Of the total possible choices of the experiment (n),

• None or More (m) are favorable to the occurrence of an event and
• The others are not favourable to the occurrence of the event.

Calculation and Symbol

The number of Unfavorable Choices or failures

• is the difference between the total number of possible choices and the number of favorable choices in the experiment.
• is a complimentary of the number of favourable choices.
• is represented by the symbol m^c

⇒ Number of Unfavourable Choices (Or) Failures

= Total Number of possible choices − Number of Favorable/Favorable choices (successes).

⇒ m_A^c = n − m_A

∴ Total Number of possible choices

	=	Number of Favorable choices + Number of Unfavorable Choices
Or	=	Number of Successes + Number of Failures

⇒ n = m_A + m_A^c

Examples

1. In the experiment of tossing a coin:

   There are two possible results or outcomes or choices i.e. there are two possible elementary events.

   1. The event of getting a HEAD
   2. The event of getting a TAIL

   Total number of possible choices = 2

   ⇒ n = 2

   Where

   V : the event of getting a TAIL

   For Event V

   Number of unfavorable/unfavourable Choices (Or) Failures

   = 1 {HEAD}

   All elementary events of the experiment where we get a HEAD are unfavorable to the occurrence of Event V

   ⇒ m_V^c = 1

   For Event V (alternative)

   Number of favorable/favourable Choices (Or) Successes

   = 1 {TAIL}

   All elementary events of the experiment where we get a TAIL are favorable to the occurrence of Event V

   ⇒ m_V = 1

   Number of Unfavourable Choices (Or) Failures

   = Total Number of possible choices − Number of Favorable/Favorable choices (Or) Successes.

   ⇒ mVc	=	n − mV
   	=	2 − 1
   	=	1

2. In the experiment of throwing a die:

   There are six possible results or outcomes or choices i.e. there are six possible elementary events.

   1. The event of 1 appearing on the face of the die
   2. The event of 2 appearing on the face of the die
   3. ...
   4. The event of 6 appearing on the face of the die

   Total number of possible choices = 6

   ⇒ n = 6

   Where

   M : the event of getting an even number

   For Event M

   Number of unfavorable/unfavourable Choices (Or) Failures

   = 3 {ONE, THREE, FIVE}

   All elementary events of the experiment where do not get an even number i.e. where we get an odd number are unfavorable to the occurrence of Event M

   ⇒ m_M^c = 1

   For Event M (alternative)

   Number of favorable/favourable Choices (Or) Successes

   = 3 {TWO, FOUR, SIX}

   All elementary events of the experiment where we get an even number are favorable to the occurrence of Event M

   ⇒ m_M = 3

   Number of Unfavourable Choices (Or) Failures

   = Total Number of possible choices − Number of Favorable/Favorable choices (Or) Successes.

   ⇒ mMc	=	n − mM
   	=	6 − 3
   	=	3

   Where

   P : the event of getting an even number

   For Event P

   Number of unfavorable/unfavourable Choices (Or) Failures

   = 4 {ONE, TWO, THREE, FOUR}

   All elementary events of the experiment where do not get do not get a number greater than 4 (or we get a number less than or equal to 4) are unfavorable to the occurrence of Event P

   ⇒ m_P^c = 4

   For Event P (alternative)

   Number of favorable/favourable Choices (Or) Successes

   = 2 {FIVE, SIX}

   All elementary events of the experiment where we get a number greater than 4 are favorable to the occurrence of Event P

   ⇒ m_P = 2

   Number of Unfavourable Choices (Or) Failures

   = Total Number of possible choices − Number of Favorable/Favorable choices (Or) Successes.

   ⇒ mPc	=	n − mP
   	=	6 − 2
   	=	4