8.01 Counting techniques

The probability of an event occurring as:

$P(event)=\frac{\text{number of favourable outcomes}}{\text{total possible outcomes}}$P(event)=number of favourable outcomestotal possible outcomes​

When calculating probabilities, using counting techniques to determine possible outcomes, makes the process easier. This chapter will focus on a number of these counting techniques.

The fundamental counting principle

Consider tossing a coin then rolling a six-sided dice. Let's look at all the possible outcomes by constructing a tree diagram. We will consider tossing the coin and rolling the dice as consecutive events.

The sample space of tossing then coin and then rolling a dice is:

$\left\{\text{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}\right\}${H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

We can see there is a total of $12$12 outcomes. We can also see from the tree diagram for each of the $2$2 possibilities for the coin toss there are $6$6 possibilities for the roll of the dice. We can calculate the total number of outcomes without determining the sample space by considering the number of outcomes of each event and multiplying them together, to account for all the possibilities, as follows:

Let's consider an application where the power of this method of counting becomes clear. In this example, it would be very inefficient to work out the number of outcomes by first determining the sample space.

Consider the number plate below. It has the format of three letters followed by three numbers.

To work out the total number of possible number plates, including those with repeated numbers and letters, we need to consider the possible outcomes for each position on the number plate.

For each letter, there are $26$26 possibilities. For each number, there are $10$10 possibilities (the numbers $0-9$0−9).

Therefore, the number of possibilities for each position is shown below:

For each of the $26$26 possibilities for the first letter, there are $26$26 for the second letter and for each of these combinations there are $26$26 possibilities for the third letter, and so on. We can multiply the possibilities for each position (we consider the selection of the letter or number of each position an event) to find the total as follows:

This is called the fundamental counting principle (or multiplication principle) and can be stated as follows:

Worked example

Example 1

A password contains six digits, which can be lower case letters or the numbers $0-9$0−9. Calculate the total number of possible passwords if each letter and number can only be used once.

Think: To use the fundamental counting principle, we need to first determine the number of possibilities for each digit. 

Do: There are $6$6 digits, with $26$26 possible letters and $10$10 possible numbers. Therefore, for each digit, there are $36$36 possibilities. However, The number of possibilities reduces by $1$1 for each subsequent digit as we can't use the same letter or number twice. Hence, the total number of possible passwords is:

Practice questions

Question 1

Question 2

Question 3