Question

Problem #3: Let A and B be two events on the sample space S. Then show that a. P(B) P(AOB)+P(AnB) b. If Bc A, then show that P(A)2 P(B) Show that P(A| B)=1-P(A|B) C. P(A) d. If A and B are mutually exclusive events then show that P(A| AUB) = PA)+P(B) Problem 4: If A and B are independent events then show that A and B are independent. If A and B are independent then show that A and B are also independent Problem # 5: A group of five undergraduate and ten graduate students are available to fill certain student government posts. If six students are randomly selected from this group, find the probability that Exactly three undergraduates among 6 will be selected At most 1 undergraduate among 6 will be selected a. b. Problem # 6: Three methods (A, B and C) are available for teaching a certain computer skill at a computer school. The failure rates are 20%, 15% and 10% for method A, B and C respectively. Among these three methods, A has been used 40%, B has been used 30% and C has been used 30% of time by the students a. What is the probability that a randomly selected student will fail? b. Suppose you have randomly selected a student from this school and found that he/she failed what is the probability that the student was taught by method A? What is the probability that the student was taught by method B? Problem # 7: Patients arriving at a hospital outpatient clinic can select one of three service stations. Suppose that physicians are assigned randomly to the stations and that the patients therefore have no station preference. Three patients arrive at the clinic and their selection of stations is observed. a. List the sample points for the experiment b. Let A be the event that each station receives a patient. Find P(A) Problem #8: An experimenter wishes to investigate the effect of three variables: temperature, pressure and type of catalyst on the yield in a refining process. If the experimenter intends to use five setting for temperature (7,), four setting for pressure (,..,P,) and three setting for catalyst (C,C,,C) How many experimental runs will have to be conducted if the experiment wishes to run all possible combinations of pressure, temperature and catalysts?? What is the probability that a randomly selected run will contain setting C,? Problem # 9: Of the items produced daily by a factory, 45% come from Plant I and 55% come from Plant II. It is known that Plant I has a defect rate of 10% where as Plant II has a defect rate of 15%. If an item is select at random from this factory, a. Find the probability that the selected item is defective b. Find the probability that the selected item is not defective

Answer 1

Solution 3

Soluth:Let A aB ar events Cans idby PCANBHANE) = PCB) b BCA ABU(BnA) event A is compased of fue disjuiht eent B ke. usiny Axidn 3 PA)= PB)Pn8 pA'n)O 6'nA je: The Now Cnsidor e PIA I8)PAne) PB) Pre)-PCANB) PCB) D fCB) PrB) PCB) 1- PCAIB)

d) Tf ARB dnt je. PCANB 7-o Chsider, PAIAUB) =P(An(AUB)) PCAUB) PCCANA)UCANB PIA)tPCB)-PAnB) PCAVCANB)) PA)HPIB-0 PCA +PCANB)-PCAM AnB) PCA)tPCB) PIA)+0-PANBJ prA)tP(B) PCAIAUB)= PrA)HPB)