Copyright © 2012 Pearson Education, Inc. All rights reserved Chapter 8 Counting Principles; Further Probability Topics

Copyright © 2012 Pearson Education, Inc. All rights reserved 8.1 The Multiplication Principle; Permutations

8 - 3 The Fundamental Counting Principle  If there are m ways to do one thing, and n ways to do another, then there are m*n ways (possible outcomes) of doing both.  The Fundamental Counting Principle is the guiding rule for finding the number of ways to accomplish two tasks. © 2012 Pearson Education, Inc.. All rights reserved.

8 - 4 Example:  How many possible outcomes are there when you want to flip a coin and roll a die ? © 2012 Pearson Education, Inc.. All rights reserved.

8 - 5  Sample Spaces  A listing of all the possible outcomes is called the sample space and is denoted by the capital letter S.  The sample space for the experiments of flipping a coin and rolling a die are S = { H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.  Sure enough, there are 12 possible ways. The fundamental counting principle allows us to figure out that there are twelve ways without having to list them all out. © 2012 Pearson Education, Inc.. All rights reserved.

8 - 6 Example: © 2012 Pearson Education, Inc.. All rights reserved. How many possible outcomes are there if you want to hit one note on a piano and play one string on a ukulele?

8 - 7 © 2012 Pearson Education, Inc.. All rights reserved. The Counting Principle also works for more than two activities.

8 - 8  Remember: The Counting Principle is easy! Simply MULTIPLY the number of ways each activity can occur. © 2012 Pearson Education, Inc.. All rights reserved.

8 - 9 Example A combination lock can be set to open to any 4-digit sequence. (a) How many sequences are possible? (b) How many sequences are possible if no digit is repeated? Solution: © 2012 Pearson Education, Inc.. All rights reserved.

ClassWork A teacher is lining up 8 students for a spelling bee. How many different line-ups are possible? Solution © 2012 Pearson Education, Inc.. All rights reserved.

© 2012 Pearson Education, Inc.. All rights reserved.

A Permutation is an ordered Combination. © 2012 Pearson Education, Inc.. All rights reserved. To help you to remember, think "Permutation... Position"

Example A teacher wishes to place 5 out of 8 different books on her shelf. How many arrangements of 5 books are possible? Solution : © 2012 Pearson Education, Inc.. All rights reserved.

Example Find the number of permutations of the letters L, M, N, O, P, and Q, if just three of the letters are to be used. Solution: © 2012 Pearson Education, Inc.. All rights reserved.

© 2012 Pearson Education, Inc.. All rights reserved.

Example: Television Panel  A televised talk show will include 4 woemn and 3 men as panelists. (a) How many ways can the panelists be seated in a row of 7 chairs? (b) In how many ways can the panelists be seated if the men and women are to be alternated? © 2012 Pearson Education, Inc.. All rights reserved.

Distinguishable Permutation If the n objects in a permutation are not all distinguishable- that is, if there are n 1 of type 1, n 2 of type 2, and so on for r different types, then the number of distinguishable permutations is © 2012 Pearson Education, Inc.. All rights reserved.

Example In how many ways can the letters in the word Tennessee be arranged? Solution: © 2012 Pearson Education, Inc.. All rights reserved.

Class Work: Mississippi © 2012 Pearson Education, Inc.. All rights reserved.  In how many ways can the letters in the work Mississippi be arranged?

Copyright © 2012 Pearson Education, Inc. All rights reserved 8.2 Combinations

Combinations and Permutation © 2012 Pearson Education, Inc.. All rights reserved. If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation. In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: So, in Mathematics we use more accurate language:

© 2012 Pearson Education, Inc.. All rights reserved. "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. "The combination to the safe is 472". Now we do care about the order. "724" won't work, nor will "247". It has to be exactly

© 2012 Pearson Education, Inc.. All rights reserved.

Example How many committees of 4 people can be formed from a group of 10 people? Solution: © 2012 Pearson Education, Inc.. All rights reserved.

Example From a class of 15 students, a group of 3 or 4 students will be selected to work on a special project. In how many ways can a group of 3 or 4 students be selected? Solution: © 2012 Pearson Education, Inc.. All rights reserved.

© 2012 Pearson Education, Inc.. All rights reserved.

Classwork (a) How many 4-digit code numbers are possible if no digits are repeated? (b) A sample of 3 light bulbs is randomly selected from a batch of 15. How many different samples are possible? (c) In a baseball conference with 8 teams, how many games must be played so that each team plays every other team exactly once? (d) In how many ways can 4 patients be assigned to 6 different hospital rooms so that each patient has a private room? © 2012 Pearson Education, Inc.. All rights reserved.

Copyright © 2012 Pearson Education, Inc. All rights reserved 8.3 Probability Applications of Counting Principles

Example The Environment Protection Agency is considering inspecting 6 plants for environmental compliance: 3 in Chicago, 2 in Los Angeles, and 1 in New York, Due to a lack of inspectors, they decide to inspect 2 plants selected at random. What is the probability that 1 New York plant and 1 Los Angeles plant are selected? © 2012 Pearson Education, Inc.. All rights reserved.

Example From a group of 22 nurses, 4 are to be selected to present a list of grievances to management. (a )In how many ways can this be done? (b) If 8 of the 22 nurses are men, what is the probability that exactly 2 men are among the 4 nurses selected? © 2012 Pearson Education, Inc.. All rights reserved.

Example When shipping diesel engines abroad, it is common to pack 12 engines in one container. Suppose that a company has received complaints from its customers that many of the engines arrive in nonworking condition. To help solve this problem, the company decides to make a spot check of containers after loading. The company will test 3 engines from a container at random; if any of the 3 are nonworking, the container will not be shipped until each engine in it is checked. Suppose a given container has 4 nonworking engines. Find the probability that the container will not be shipped. © 2012 Pearson Education, Inc.. All rights reserved.

Exmaple  In a club with 9 make and 11 female members, a 5-mumber committee will be randomly chosen. Find the probability that the committee contains the following: a. All men  b. All women. c. 3 men and 2 women. d. 2 men and 3 women e. At least 4 women f. No more than 2 men. © 2012 Pearson Education, Inc.. All rights reserved.

Classwork  Find the probability that the 2-card hand described above contains the following: a. Only hearts b. At least 3 aces c. All spades d. 2 cards of the same suit e. Only face cards f. No face cards G.No card higher than 8 (count ace as 1) © 2012 Pearson Education, Inc.. All rights reserved.

Classwork  A bridge hand is made up of 13 cards from a deck of 52. Find the probabilities that a hand chosen at random contains the following:  a. Only hearts b. At least 3 aces c. Exactly 2 aces and exactly 2 kings d. 6 of one unit, 4 of another, and 3 of another. © 2012 Pearson Education, Inc.. All rights reserved.

Copyright © 2012 Pearson Education, Inc. All rights reserved 8.4 Binomial Probability

© 2012 Pearson Education, Inc.. All rights reserved.

© 2012 Pearson Education, Inc.. All rights reserved.

Example Find the probability of getting exactly 4 heads in 8 tosses of a fair coin. © 2012 Pearson Education, Inc.. All rights reserved.

Example Assuming that selection of items for a sample can be treated as independent trials, and that the probability that any 1 item is defective is 0.01, find the probability of 2 or 3 defective items in a random sample of 15 items from a production line. © 2012 Pearson Education, Inc.. All rights reserved.

© 2012 Pearson Education, Inc.. All rights reserved.

Example  Suppose that a family has 5 children. Also, suppose that the probability of having a girl is ½. Find the probabilities that the family has the following children.  Exactly 2 girls and 3 boys.  Exactly 3 girls and 2 boys.  No girls.  At least 4 girls.  No more than 4 girls. © 2012 Pearson Education, Inc.. All rights reserved.

Classwork  A die is rolled 12 times. Find the probabilities of rolling the following:  1)Exactly 12 ones.  2)Exactly 6 ones.  3)No more than 3 ones.  4)No more that 1 ones.  5)At least 2 ones.  6)At least 10 ones. © 2012 Pearson Education, Inc.. All rights reserved.

Copyright © 2012 Pearson Education, Inc. All rights reserved Probability Distributions; Expected Value 8.5

© 2012 Pearson Education, Inc.. All rights reserved. In other words, a random variable is a set of possible values from a random experiment.

© 2012 Pearson Education, Inc.. All rights reserved. Example Tossing a coin: we could get Heads or Tails.

Example Throw a die once Random Variable X = "The score shown on the top face". X could be 1, 2, 3, 4, 5 or 6 So the Sample Space is {1, 2, 3, 4, 5, 6}  © 2012 Pearson Education, Inc.. All rights reserved.

Probability Distribution  Aprobability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. © 2012 Pearson Education, Inc.. All rights reserved.

© 2012 Pearson Education, Inc.. All rights reserved. So: (1) We have an experiment (such as tossing a coin) (2) We give values to each event (3) The set of values is a Random Variable

Example A shipment of 12 computer monitors contains 3 broken monitors. A shipping manager checks a sample of two monitors to see if any are broken. Find the probability distribution for the number of broken monitors. © 2012 Pearson Education, Inc.. All rights reserved.

Example Find the probability distribution and draw a histogram for the number of tails showing when three coins are tossed. © 2012 Pearson Education, Inc.. All rights reserved.

© 2012 Pearson Education, Inc.. All rights reserved.

Example When you roll a die, you will be paid $1 for odd number and $2 for even number. (1)Construct a probability distribution for the money that you would be paid. (2) Find the expected value of money you get for one roll of the die. © 2012 Pearson Education, Inc.. All rights reserved.

Example  Suppose 3 marbles are drawn from a bag containing 3 yellow and 4 white marbles. a. Draw a histogram for the number of yellow marbles in the sample. b. What is the expected number of yellow marbles in the sample. © 2012 Pearson Education, Inc.. All rights reserved.

Class Work  Suppose a die is rolled 4 times. a. Find the probability distribution for the number of times 1 is rolled. b. What is the expected number of times 1 is rolled? © 2012 Pearson Education, Inc.. All rights reserved.

© 2012 Pearson Education, Inc.. All rights reserved.

Example  Suppose a family has 3 children.  (a) Find the probability distribution for the number of girls. (b)Find the expected number of girls in a 3- child family using the distribution form part (a) and compare your result with the formula above. © 2012 Pearson Education, Inc.. All rights reserved.