Probability : Fundamental Counting Principle (Theorem) of Addition

Fundamental Counting Principle of Addition

If a total event can be accomplished in two or more mutually exclusive alternative events/ways, then the number of ways in which the total event can be accomplished is given by the sum of the number of ways in which each alternative-event can be accomplished.

Number of ways in which the total event can be accomplished

= (Number of ways in which the first alternative-event can be accomplished)
  + (Number of ways in which the second alternative-event can be accomplished)
  + (Number of ways in which the third alternative-event can be accomplished)
  + ....
  + ....

⇒ nE = nEa + nEb + nEc + ....

Fundamental Counting Principle/Theorem of Addition : Illustration ยป 1

In how many ways can a committee 4 members be chosen from a group of 6 men and 5 women such that the committee consists of at least 2 women?

Total number of members = 6 men + 5 women
= 11

Number of Members selected = 4.

Experiment : Choosing 4 members

Let "E" be the event of choosing the committee with at least 2 women in it.

• For Event E

Event E can be accomplished in three alternative ways
Choosing the Committee with 4 members
Ea » Choosing the Committee with 2 Women and 2 Men
Eb » Choosing the Committee with 3 Women and 1 Man
Ec » Choosing the Committee with 4 Women and 0 Men
  • Total Event (E) = Choosing the committee of 4 members with at least 2 women in it.
  • 1st alternative-event (Ea) = Choosing the committee with 2 women and 2 men
  • 2nd alternative-event (Eb) = Choosing the committee with 3 women and 1 man
  • 3rd alternative-event (Ec) = Choosing the committee with 4 women and 0 men

[Ea, Eb, Ec are Mutually Exclusive Events. Occurrence of one of these events prevents the occurrence of the others. If the committee is chosen in one of these ways we can say that it was not chosen in the other ways]

» Ea

The number of ways in which the committee can be chosen with 2 women and 2 men = 150
n(Ea) = 150

Working Notes :

Considering the event of choosing the 2 women and 2 men to be the total event.

The total event can be sub-divided into two independent sub-events

  • Total Event (Ea) = Choosing the 2 women and 2 men
  • 1st sub-event (Ea1) = Choosing the 2 women from the available 6
  • 2nd sub-event (Ea2) = Choosing the 2 men from the available 5
Working Table
Women × Men
Available 6 5
To Choose 2 2
Choices 6C2 5C2


Fundamental Counting Theorem (of Multiplication): Where an event can be sub divided into two or more independent sub-events, the total number of ways in which the total event can be accomplished is equal to the product of the number of ways in which the sub-events can be accomplished.

By the fundamental counting theorem of multiplication, nEa = nEa1 × nEa2
= 6C2 × 5C2 +
=
6 × 5
2 × 1
 ×
5 × 4
2 × 1
= 15 × 10
= 150

» Eb

The number of ways in which the committee can be chosen with 3 women and 1 man = 100
n(Eb) = 150

Working Notes :

Considering the event of choosing the 3 women and 1 man to be the total event.

The total event can be sub-divided into two independent sub-events

  • Total Event (Eb) = Choosing the 3 women and 1 man
  • 1st sub-event (Eb1) = Choosing the 3 women from the available 6
  • 2nd sub-event (Eb2) = Choosing the 1 man from the available 5
Working Table
Women × Men
Available 6 5
To Choose 3 1
Choices 6C3 5C1


Fundamental Counting Theorem (of Multiplication): Where an event can be sub divided into two or more independent sub-events, the total number of ways in which the total event can be accomplished is equal to the product of the number of ways in which the sub-events can be accomplished.

By the fundamental counting theorem of multiplication, nEb = nEb1 × nEb2
= 6C3 × 5C1
=
6 × 5 × 4
3 × 2 × 1
 ×
5
1
= 20 × 5
= 100

» Ec

The number of ways in which the committee can be chosen with 4 women and 0 men = 15
n(Ec) = 15

Working Notes :

Considering the event of choosing the 4 women and 0 men to be the total event.

The total event can be sub-divided into two independent sub-events

  • Total Event (Ec) = Choosing the 4 women and 0 men
  • 1st sub-event (Ec1) = Choosing the 4 women from the available 6
  • 2nd sub-event (Ec2) = Choosing the 0 men from the available 5
Working Table
Women × Men
Available 6 5
To Choose 4 0
Choices 6C4 5C0


Fundamental Counting Theorem (of Multiplication): Where an event can be sub divided into two or more independent sub-events, the total number of ways in which the total event can be accomplished is equal to the product of the number of ways in which the sub-events can be accomplished.

By the fundamental counting theorem of multiplication, nEc = nEc1 × nEc2
= 6C4 × 5C0
=
6 × 5 × 4 × 3
4 × 3 × 2 × 1
 × 1
= 15 × 1
= 15

Therefore,

The number of ways in which the committee can be chosen
= [No. of ways in which the committee with 2 women and 2 men (1st alternative event) can be chosen]
  + [No. of ways in which the committee with 3 women and 1 man (2nd alternative-event) can be chosen]
  + [No. of ways in which the committee with 4 women and 0 men (3rd alternative-event) can be chosen]

⇒ nE = nEa + nEb + nEc
= 150 + 100 + 15
= 265

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