Bayes stuff Red Cross and Blood Example 42% of the workers at Motor Works are female, while 67% of the workers at City Bank are female. If one of these companies is selected at random (assume a 50-50 chance for each), and then a worker is selected at random, what is the probability that the worker is female, if we know that the worker comes from City Bank? 14) Two shipments of components were received by a factory and stored in two separate bins. Shipment I has 2% of its contents defective, while shipment II has 5% of its contents defective. If it is equally likely an employee will go to either bin and select a component randomly, what is the probability that a defective component came from shipment II? 1.1-2 15) A person must select one of three boxes, each filled with clocks. The probability of box A being selected is 0.31, of box B being selected is 0.13, and of box C being selected is 0.56. The probability of finding a red clock in box A is 0.2, in box B is 0.4, and in box C is 0.9. A box is selected. Given that the box contains a red clock, what is the probability that box A was chosen? 16) In one town, 8% of 18-29 year olds own a house, as do 28% of 30-50 year olds and 52% of those over 50. According to a recent census taken in the town, 26.9% of adults in the town are 18-29 years old, 36.0% are 30-50 years old, and 37.1% are over 50. What percentage of house-owners are 30-50 years old? 1.1-3 1
17) Among students at one college are 3934 women and 3103 men. The following table provides relative-frequency distributions for subject major for males and females at the college. Major Relative frequency for women Relative frequency for men Humanities 0.198 0.173 Science 0.303 0.364 Social Science 0.166 0.349 Other 0.166 0.114 A student is selected at random from the college. Determine the probability that the student selected is female given that he or she is a Humanities major. 1.1-4 18) The incidence of a certain disease in the town of Springwell is 4%. A new test has been developed to diagnose the disease. Using this test, 91% of those who have the disease test positive while 6% of those who do not have the disease test positive ("false positive"). If a person tests positive, what is the probability that he or she actually has the disease? 1.1-5 Section 12.8 The Counting Principle and Permutations 2
Counting Principle If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct ways, then the two experiments in that specific order can be performed in M N distinct ways. 12.8-7 A password is to consist of two lower case letters followed by four digits. Determine how many different passwords are possible if a) repetition of letters and digits is permitted. b) repetition of letters and digits is not permitted. c) the first letter must be a vowel, and the first digit cannot be a 0, and repetition of letters and digits is not permitted. 12.8-8 13 people, need 3 on a committee. They will be president, VP, and secretary. How many committees? Now all 13 on committee, again each with jobs. 1.1-9 3
Example 3: Cell Phones In how many different ways can six different cell phones be arranged on top of one another? Consider the five letters a, b, c, d, e. In how many distinct ways can three letters be selected and arranged if repetition is not allowed? In how many different ways can the letters of the word BLINN be arranged? TALLAHASSEE MISSISSIPPI 12.8-10 12.8-11 Seating Problems: We have 5 boys and 5 girls. In how many way can we seat them in a row if: 1) No restrictions 2) Alternate gender 3) Boys all together 1.1-12 4
We have 5 boys and 5 girls continued: Girls all together A girl on each end A boy on one end and a girl on the other NOT all of the girls next to each other 1.1-13 Section 12.9 Combinations 13 people, need 3 on a committee. How many committees? Now all 13 on committee. 1.1-15 5
7 red, 8 green marbles. We draw 4. How many ways to get: Exactly 1 red? Exactly 1 green? At least 2 red? 1.1-16 7 red, 8 green, and 9 blue marbles. We draw 4. How many ways to get: Exactly 1 red? Exactly 1 green? At least 2 red? 1.1-17 7 red, 8 green, and 9 blue marbles. We draw 4. How many ways to get: Exactly 2 red: Exactly 2 green: Exactly 2 red and exactly 2 green: Exactly 2 red or exactly 2 green: 1.1-18 6
Example 2: Museum Selection While visiting New York City, the Friedmans are interested in visiting 8 museums but have time to visit only 3. In how many ways can the Friedmans select 3 of the 8 museums to visit? Example 3: Floral Arrangements Jan Funkhauser has 10 different cut flowers from which she will choose 6 to use in a floral arrangement. How many different ways can she do so? Example 4: Dinner Combinations At the Royal Dynasty Chinese restaurant, dinner for eight people consists of 3 items from column A, 4 items from column B, and 3 items from column C. If columns A, B, and C have 5, 7, and 6 items, respectively, how many different dinner combinations are possible? 12.9-19 12.9-20 12.9-21 7
How many 5 card hands: Total? Have at least 3 Kings? Have at least 3 spades? Have at least 1 heart? Have 3 spades and 2 hearts? Have 1.1-22 3 of one suit and 2 of another? Section 12.10 Solving Probability Problems by Using Combinations Example 1: Committee of Three Women A club consists of four men and five women. Three members are to be selected at random to form a committee. What is the probability that the committee will consist of three women? 12.10-24 8
25 Example 5: Rare Coins Conner Shanahan s rare coin collection is made up of 8 silver dollars, 7 quarters, and 5 dimes. Conner plans to sell 8 of his 20 coins to finance part of his college education. If he selects the coins at random, what is the probability that 3 silver dollars, 2 quarters, and 3 dimes are selected? A temporary agency has six men and five women who wish to be assigned for the day. One employer has requested four employees for security guard positions, and the second employer has requested three employees for moving furniture in an office building. If we assume that each of the potential employees has the same chance of being selected and being assigned at random and that only seven employees will be assigned, find the probability that a) three men will be selected for moving furniture. b) three men will be selected for moving furniture and four women will be selected for security guard positions. 12.10-12.10-26 Section 12.11 Binomial Probability Formula 9
To Use the Binomial Probability Formula There are n repeated independent trials. Each trial has two possible outcomes, success and failure. For each trial, the probability of success (and failure) remains the same. Binomial Probability Formula The probability of obtaining exactly x successes, P(x), in n independent trials is given by: P( x)= ( n C x )p x q n x where p is the probability of success on a single trial and q (= 1 p) is the probability of failure on a single trial. A basket contains 3 balls: 1 red, 1 blue, and 1 yellow. Three balls are going to be selected from the basket. Find the probability that a. no red balls are selected if we draw without replacement b. no red balls are selected if we draw with replacement 12.11-28 12.11-29 12.11-30 10
A basket contains 1 red, 1 blue, and 1 yellow. Three balls are going to be selected from the basket with replacement. Find the probability: exactly 1 red ball is selected. exactly 2 red balls are selected. exactly 3 red balls are selected. Example 4: Planting Trees The probability that a tree planted by a landscaping company will survive is 0.8. If they plant four determine the probability that a) none will survive. b) at least one will survive. c) At least two will survive. 12.11-31 12.11-32 The Dr. Pepper says 1 in 8 wins. If you buy 20 of those drinks this semester, find the probability (A) you win at least twice: (B) You do not win at all: (C) You win fewer than three times: (D) For the class, how many of A,B,C would we expect? 11
In one city, the probability that a person will pass his or her driving test on the first attempt is 0.68. 19 people are selected at random from among those taking their driving test for the first time. What is the probability that: The number passing is less than 4? The number passing is more than 10? The number passing is between 2 and 8 (inclusive)? 41% of the murder trials in one district result in a guilty verdict. Five murder trials are selected at random from the district. Determine the probability distribution of X, the number of trials among the five selected in which the defendant is found guilty. A multiple choice test consists of four questions. Each question has five possible answers of which only one is correct. A student guesses on every question. Find the probability distribution of X, the number of questions she answers correctly. Copyrig ht 12