CHAPTER 4: PROBABILITY CONCEPTS 4.8 Counting Rules

BASIC COUNTING RULE  Tree Diagrams:  Example: A home builder in Arizona offers four models of homes—the Shalimar, Palacia, Valencia, and Monterey—each in three different elevations. How many choices are there for the selection of a home, including both model and elevation?

BASIC COUNTING RULE  Suppose r actions are to be performed in a definite order  M1 possibilities for the first action, m2 for the second action, etc.  To get the total number of possibilities: m1 x m2 x …

EXAMPLE 4.28  The license plates of Arizona consist of three letters followed by three digits.  A. How many different license plates are possible?  B. How many possibilities are there for license plates on which no letter or digit is repeated?

FACTORIALS  The product of the first k positive integers is called k factorial : k! = k(K – 1)…2 · 1 Also, 0! = 1

EXAMPLE 4.29  Determine:  3!  4!  5!

PERMUTATIONS  r objects from a collection of m objects is any ordered arrangement of r of the m objects

EXAMPLE 4.31  In an exacta wager at the race track, a bettor picks the two horses that she thinks will finish first and second in a specified order. For a race with 12 entrants, determine the number of possible exacta wagers.

EXAMPLE 4.32  A student has 10 books to arrange on a shelf of a bookcase. In how many ways can the 10 books be arranged?

COMBINATIONS  r objects from a collection of m objects where order doesn’t matter

EXAMPLE 4.33  Consider the collection consisting of the five letters a, b, c, d, and e:  A. List all possible combinations of three letters from this collection of five letters.  B. use part (a) to determine the number of possible combinations of three letters that can be formed from the collection of five letters.

EXAMPLE 4.34  To recruit new members, a CD club advertises a special introductory offer: a new member agrees to buy 1 CD at regular club prices and receives free any 4 CDs of his choice from a collection of 69 CDs. How many possibilities does the new member have for the selection of the 4 free CDs?

NUMBER OF POSSIBLE SAMPLES  The number of possible samples of size n from a population of size N is:

EXAMPLE 4.35  An economics professor is using a new method to teach a junior-level course with an enrollment of 42 students. The professor wants to conduct in-depth interviews with the students to get feedback on the new teaching method, but she does not want to interview all of them. She decides to interview a sample of 5 students from the class. How many different samples are possible?

EXAMPLE 4.36  The quality assurance engineer of a television manufacturer inspects TVs in lots of 100. He selects 5 of the 100 TVs at random and inspects them thoroughly. Assuming that 6 of the 100 TVs in the current lot are defective, find the probability that exactly 2 of the 5 TVs selected by the engineer are defective.

HOMEWORK  P. 206 