Probability Theory

Counting

A theoretical approach to determine probabilities requires the ability to count the number of ways certain events can occur. In some cases, counting is simple because there is only one way for an event to occur. For example, there is only one way in which a 4 will show up on a single roll of a die. In most cases, however, counting is not always an easy matter. Imagine trying to count the number of ways of being dealt a pair in 5 card poker.

The fundamental principle of counting is often used when many selections are made from the same set of objects. Suppose we want to know the number of different ways four people can line up in a carnival line. The first spot in line can be occupied by any of the four people. The second can be occupied any of the three people who are left. The third spot can be filled by either of the two remaining people, and the fourth spot is filled by the last person. So, the total number of ways four people can create a line is equal to the product 4 × 3 × 2 × 1 = 24. This product can be abbreviated as 4! (read "4 factorial"). In general, the product of the positive integers from 1 to n can be denoted by n! which equals n × (n-1) × (n-2) ×...2 × 1. It should be noted that 0! is by definition equal to 1.

The example of the carnival line given above illustrates a situation involving permutations. A permutation is any arrangement of n objects in a definite order. Generally, the number of permutations of n objects is n. Now, suppose we want to make a line using only two of the four people. In this case, any of the four people can occupy the first space and any of the three remaining people can occupy the second space. Therefore, the number of possible arrangements, or permutations, of two people from a group of four, denoted as P4,2 is equal to 4 × 3 = 12. In general, the number of permutations of n objects taken r at a time is

This can be written more compactly as Pn,r = n!/(n-r)!

Many times the order in which objects are selected from a group does not matter. For instance, we may want to know how many different 3 person clubs can be formed from a student body of 125. By using permutations, some of the clubs will have the same people, just arranged in a different order. We only want to count then number of clubs that have different people. In these cases, when order is not important, we use what is known as a combination. In general, the number of combinations denoted as Cn,r or is equal to Pn,r /r! or Cn,r = n!/r! × (n-r)! For our club example, the number of different three person clubs that can be formed from a student body of 125 is C125,3 or 125!/3! × 122! = 317,750.