Math 141-copyright Joe Kahlig, 10C Page 1Section 7.5: Conditional Probability a nd Independenceand Section 7.6: Bayes’ TheoremDefinition: Conditional probability is the probability of an event, A, will occur given thatanother event, B, has already happened. This is denoted as P (A|B).Example: This table classifies the English, History, Math, and Poly Sci majors at State U accordingto their year. (There are no dou ble majors.)Freshmen(F) Sophomores(Soph) Juniors(J) Seniors(Sr) TotalsEnglish(E) 64 35 31 41 171History(H) 55 41 33 52 181Math(M) 29 32 50 69 180Poly Sci(PS) 70 33 41 37 181Totals 218 141 155 199 713If a student is selected at random, findA) P(H | soph) =B) P( (soph ∪ J) |PS) =Interpret these Statements: Express these questions using the correct probability notation.A factory makes widgets on three different machines; A, B and C; and a widget is selected at random.1. What is the probability that it is defective knowing that the w idget was made on machine C?2. If the wid get was made on machine A, what is the probability that it is defective?3. What is the probability it is mad e on machine A and is not d efective?4. What is the probability the wid get is selected from line B or is defective?5. What is the probability that the defective widget was made on machine B?6. What is the probability that the widget is defective and came from machine A or machine B?Math 141-copyright Joe Kahlig, 10C Page 2Example: Use th e pr obab ility distribution and the events to answer these questions.Sa b c d e fprob 0.15 0.08 0.21 0.12 0.25 0.19E = { a, c, d, e} F = {b, d, f} G = {a, b, d}ComputeA) P (E|G) =B) P (G|F ) =Definition: Let E and F be two events of the sample space S. Then the conditional probability thatE occurs given that F has occurred is defined asP (E|F ) =Example: It is know from a survey that 29% of the people buy product A, 36% of the people buyproduct B and 11% buy both prod ucts. Find the probability thatA) the person buys product A if they bought product B.B) the person does not buy product B knowingthat they bought product A.Math 141-copyright Joe Kahlig, 10C Page 3Example: A Jar has 5 red, 4 green, and 3 yellow items. Two items are drawn in succession withoutreplacement. Construct the prob ability tree that represents this experiment.Example: Draw the tree that represents this experiment: Draw one ball fr om box A and place it intobox B. Then draw one ball from box B.Box ABox A3 red 2 red5 green 5 yellowMath 141-copyright Joe Kahlig, 10C Page 4Example: You have two jars. Jar 1 h ad 4 red, 3 green and 2 black balls. Jar 2 has 8 red and 5 greenitems. The experiment is to draw a single ball. We have been told that a person is twice as likely todraw an item from Jar 1 as from Jar 2.A) Draw the probability tree for this pr ob lem.B) P (r|J1) =C) P (g|J2) =D) P (b|J2) =E) P (J2 ∩ r) =F) P (g) =Math 141-copyright Joe Kahlig, 10C Page 5Example: Use fact that P (C ∩ D) = 0.09 and the tree to answer these questions.A) P (E|A) =B) P (B|D) =C) P (A ∪ D) =EDDDEEBAC0.80.70.40.1Example: Based on data from a dental survey, it has been determined that 42% of 12-year olds havenever had a cavity, 34% of 13-year olds have never had a cavity, and 28% of 14-year olds have neverhad a cavity. A child is selected at random from a group of s tu d ents of which 28% of them are 12-yearolds, 45% of them are 13-year olds and the rest are 14-year olds.A) What is the probability thatthe selected student has a cavity?B) What is the probability that the student,who had a cavity, was 14 years old?Math 141-copyright Joe Kahlig, 10C Page 6Example: A medical test has been designed to detect the presence of a certain condition. Amongthose who have the condition, the probability that the test will detect it is 98%. The test will give afalse positive 6% of the time. It is generally known that 12% of the population has the condition.A) If the test administered to a person is positive,what it the probability that the person actuallyhas the condition?B) If the test is administered a second time andit comes back positive both times, w hat is th eprobability that the person has the condition?Example: Draw cards from a deck of cards without replacement.A) What is the probability that thesecond card drawn is a heart.B) P(3rd card is a heart) =C) P(28th card is a heart) =D) P(1st heart | 2nd is a club) =Math 141-copyright Joe Kahlig, 10C Page 7Example: A computer store shipped 5 defective computers in its s hipment of 30 computes to a localschool. The computers are arranged on desks insid e the classroom. Find the probability that thecomputer on desk 5 is defective if the computer on d esk 1 and desk 10 are both good and the one ondesk 25 is defective.Definition: Events E an d F are said to be in dependent ifP (E|F ) = P (E) and P (F |E) = P (F )Example: A kennel raises purebred dogs. Several litters from one dog produced 16 puppies with thefollowing markings:Five had a w hite mark only on the head.three had a white mark only on the forelegs.three had a white mark on both head and forelegsThe rest had neither mark.Determine whether the events “white mark on the head” and “white m ark on the forelegs” are inde-pendent or not.Math 141-copyright Joe Kahlig, 10C Page 8Example: A kidney transplant doctor has three patients, A, B, and C, that will be receiving a kidneytransplant. The doctor has determined that patient A has a 12% chance of rejecting the kidney,patient B has a 8% chance of rejecting the kidney, and patient C has a 21% chance of rejecting thekidney.A) What is the probability that noneof the patients reject the kidney.B) What is the probability that exactlyone of the patients will reject the kidney.Example: If P (A) = 0.4 and P (B) = 0.6, what is P (A ∩ B)?Example: A retailer receives a shipm ent of TV sets from two different companies. The first shipment,from company A, is known historically to be 7.5% defective. The second, from company B, is knownto be 13% defective. If one item is selected from each shipment, what is the probability of selectingone good TV and one defective
View Full Document
Unlocking...