Chapter 3

1. Chapter 3 Probability

2. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 3-4 A. Sometimes we want to determine how many ways we can choose and arrange not a whole set, but a specified number of members from that set. 1) The number of ways that r (being a whole number which is less than or equal to n) elements out of a set of n elements can be arranged is called a permutation of n objects using r. a) This is denoted as nPr where n gives the total number in the set and r gives how many you are using. 1. The formula for a permutation is: = where . a. The symbol ! stands for a factorial, and means that we will take that number times every number below it all the way to 1. 1) For example, 6! means 6*5*4*3*2*1. 2) To calculate a factorial on the TI-83, enter the number that you want the factorial of then press the MATH key. Arrow over to the PRB menu. The 4th item on that list is the ! symbol, so enter the number 4, and then press Enter. a) Find 7! – Enter 7, MATH, PRB, 4, Enter. The screen will tell you that the answer is 5040.

3. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 3-4 A. Sometimes we want to determine how many ways we can choose and arrange not a whole set, but a specified number of members from that set. 1) The number of ways that r (being a whole number which is less than or equal to n) elements out of a set of n elements can be arranged is called a permutation of n objects using r. a) This is denoted as nPr where n gives the total number in the set and r gives how many you are using. 1. The formula for a permutation is: = where . 2. To calculate a permutation using the TI-83, enter the n value first, then press MATH. The permutation option is the second one on the menu, so press 2. Now enter the r value and then press Enter. a. Find 6P3 – Enter 6, MATH, PRB, 2, 3, Enter. The answer should be 120.

4. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 3-4 2) A special type of permutation is called a distinguishable permutation. a) These occur when you want to order a group of objects in which some of the objects are the same. 1. For example, you want to order a group of letters that includes four A’s, two B’s, and one C. 2. The formula for distinguishable permutations is: where n1 is of one type, n2 is of another, and so on. b) Let’s say a building contractor is planning to develop a subdivision. The subdivision is to consist of 6 one-story houses, 4 two-story houses, and 2 split-level houses. 1. In how many distinguishable ways can the houses be arranged? a. There are to be 12 houses in the subdivision, 6 of which are of one type (one-story), 4 of another type (two-story), and 2 of a third type (split-level). = 1) There are 13,860 distinguishable ways to arrange those 12 houses. c) As far as I know, the TI-83 does not calculate distinguishable permutations. You will have to set it up and use the factorial feature.

5. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 3-4 B. There are also situations for which we wish to determine the number of ways an r- member subset can be formed from a set with n numbers. 1) In this situation, the order of selection doesn’t matter. a) Key words to look for: subset, committee, group. 2) This is called a combination of n elements using r and is denoted nCr. a) There are 10 people in a club. We wish to know how many ways we can select a committee (where order does not matter) consisting of 4 people. 1. The formula for a combination is: = where . 2. n = 10, r = 4. a. 10C4 = 10!/4!(10-4)! = 10!/(4!6!) = 210. 3) To calculate a combination on the TI-83, enter the n value first, then press MATH. The combination option is the third one on the menu, so press 3. Now enter the r value and then press Enter. a) Find 6C3 – Enter 6, MATH, PRB, 3, 3, Enter. The answer should be 20.

6. PROBABILITY AND STATISTICS SECTION 3-4 EXAMPLES Calculate the number of ways each of the following can be done. Then answer the corresponding probability question. A. Your company routes all materials from NY to Chicago, then from Chicago to Denver and then from Denver to L.A. There are 4 routes from NY to Chicago, 6 routes from Chicago to Denver and 3 routes from Denver to L.A. How many possible routes are there from NY to L.A.? What is the probability that one specified route is taken? Use the multiplication principle. Since there are 4 ways to perform the first action, and 6 ways to perform the second, and 3 ways to perform the third, there are (4)(6)(3) = 72 possible routes. The probability that any one specified route is taken is 1/72 = .0139

7. PROBABILITY AND STATISTICS SECTION 3-4 EXAMPLES Calculate the number of ways each of the following can be done. Then answer the corresponding probability question. You are supervisor of nurses and have 12 nurses working for you. In how many ways can you choose 4 to have the weekend off? What is the probability that 4 specific individuals will have the weekend off? In this problem, the order does not matter, so we will use a combination. n = 12 and r = 4. 12C4= 12!/4!(12-4)! = 12!/4!8! = 495 The probability that 4 specific individuals will all have the same weekend off is 1/495 = .0020

8. PROBABILITY AND STATISTICS SECTION 3-4 EXAMPLES Calculate the number of ways each of the following can be done. Then answer the corresponding probability question. You are still in charge of 12 nurses (congratulations!) and must assign one to be head of the first floor ward, one to be head of the second floor ward, one for the third floor, and one for the fourth. In how many ways can this be done? What is the probability that the assignments are made in one specified order? This is a permutation, since it matters which floor they are on. n= 12, r = 4 12P4 = 12!/(12-4)! = 12!/8! = 11,880 The probability that the assignments are made in one specified order is 1/11,880 = .00008.