Probability: A Lively Introduction

Many probability problems require counting techniques. In particular, these techniques are extremely useful for computing probabilities in a chance experiment in which all possible outcomes are equally likely. In such experiments, one needs effective methods to count the number of outcomes in any specific event. In counting problems, it is important to know whether the order in which the elements are counted is relevant or not.

In the discussion below, we use the fundamental principle of counting: if there are a ways to do one activity and b ways to do another activity, then there are a × b ways of doing both. As an example, suppose that you go to a restaurant to get some breakfast. The menu says pancakes, waffles, or fried eggs, while for a drink you can choose between juice, coffee, tea, and hot chocolate. Then the total number of different choices of food and drink is 3 × 4 = 12. As another example, how many different license plates are possible when the license plate displays a nonzero digit, followed by three letters, followed by three digits? The answer is 9×26×26×26×10×10× 10 = 158,184,000 license plates.

Permutations

How many different ways can you arrange a number of different objects such as letters or numbers? For example, what is the number of different ways that the three letters A, B, and C can be arranged? By writing out all the possibilities ABC, ACB, BAC, BCA, CAB, and CBA, you can see that the total number is 6. This brute-force method of writing down all the possibilities and counting them is naturally not practical when the number of possibilities gets large, for example the number of different ways to arrange the 26 letters of the alphabet. You can also determine that the three letters A, B, and C can be written down in six different ways by reasoning as follows. For the first position there are three available letters to choose from, for the second position there are two letters over to choose from, and only one letter for the third position. Therefore the total number of possibilities is 3 × 2 × 1 = 6. The general rule should now be evident. Suppose that you have n distinguishable objects.