An efficient way of counting is necessary to handle large masses of statistical data (e.g. the level of inventory at the end of a given month, or the number of production runs on a given machine in a 24 hour period, etc.), and for an understanding of probability.

Addition rule and Multiplication rule are techniques that will enable us to count:

• The number of ways,
• The number of samples, or
• The number of outcomes.

Suppose we have an event E, defined as

E = "day of the week"

We can write the "number of outcomes of event E" as n (E). The outcomes of event E can be Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday.

So in the example,

n (E) = 7

Addition Rule

Let E1 and E2 be mutually exclusive events (i.e. there are no common outcomes).

Let event E describe the situation where either event E1 or event E2 will occur.

The number of times event E will occur can be given by the expression:

n (E) = n (E1) + n (E2)

For example:

Consider a set of numbers S = {-4, -2, 1, 3, 5, 6, 7, 8, 9, 10}

Let the events E1, E2 and E3 be defined as:

E = choosing a negative or an odd number from S;

E1= choosing a negative number from S;

E2 = choosing an odd number from S.

Find n(E).

Since E1 and E2 are mutually exclusive events (i.e. no common outcomes):

n(E) = n(E1) + n(E2) = 2 + 5 = 7

Multiplication Rule

Suppose that event E1 can result in n(E1) possible outcomes; and for each outcome of the event E1, there are n(E2) possible outcomes of event E2.

Then in total, there will be n(E1) × n(E2) possible outcomes of the two events.

In other words, if event E is the event that both E1 and E2 must occur, then

n(E) = n(E1) × n(E2)

Suppose the only clean clothes you've got are 2 t-shirts (white and blue), and 4 pairs of jeans (small, medium, large, x-large). How many different combinations can you choose?

We can think of it as follows:

There are 2 outcomes of t-shirt choosing: white and blue;

For each t-shirt outcome, there are 4 outcomes of jeans choosing: small, medium, large, x-large.

Therefore, there will be 2 × 4 = 8 outcomes.