Question

discrete maths.

4. Give a complete solution to the following problems using the framework of Polya's four steps in problem solving. Interpret and justify your results as far as possible. (a) A manufacturer buys large quantity of products from a producer. At delivery the manufacturer makes a quality control, testing 10 randomly chosen products. If the number of bad products is 2 or more, delivery is returned to the producer. From previous tests, the producer has observed 4% of the products are bad. What is the producers risk, i.e. what is the probability that the delivery is returned to the produ- cer? (3p) (b) Are there numbers a and b such that the equality s. 1 (k + 1)(k +2) nta n + b holds for every integer n > 0? (3p)

Answer 1

understanding te publen a) Gover that we are testing to a natuly dose products of the number of bad products are two or une delivery is retired Devine a plan Tet & be a rudan variable that counts the number of bad products in a random sample of to products Carrying out the plan let è be probability (frucer) getting a bad product at is given that p=442 0.04 then q = probalability of getting a good product q=1p=1-0.04 -0.96 The follows Bihanial distribution 'with parameters na 10, p=004 Pax=r)= 104.pt.quv . 10gr @04346.963am We are asked to find the probability that delivery is returned - If number of had products are for more ie we have to find P(x2,2) pleza) = x29) = 1- (Pekzo) € P GE=)) = tefte teoretos, a =1-{17, 60486741° "c, 6043' 69619] = 1 [0.6648 +6.0.2470] M{X72) > 0.006214 so probability and that delivery is returned is 0.06214 È ikeillee) Het Kuo Scanned with . CamScanner

at (t+1) (62) be bevising aplan first we write the partial fractione for and then sum up to get the result Execting the play kerckeze ale to Executing te plan kt 1) (642) (kt =10 -*+ *.... ... k-44ế - - 1€! FNEN È tuisceran a lehe *AEH kel) CK2) n42 The tage numbers and a and b are 1, 2 respectively d) Given that 5 independent Bernouli trials are conducted with success probability we have to find the conditional probability that exactly fur heads can given that result of the first trial was success. Scanned with

There will be 16 outcomes of ahen performing a Bernali experiment five times in which 1st trial is success of the i6 aut comes we have to hand those wtcomes in which remaining accesses are seleze we get 3 svecences act of cremaining trials, hege are only a posebility PPCS)=P 7 SSSSE SSSES, SSESS, SAS SS L PCIe=lap | Treative required probabilty = P(SSSSP) MSSSES)tplSSESS)ep(sesss) = ptap) + ptc-p) +pAlp) +P$(1-P) il probability of getting a = 4 P4Cl-p) Successes giver first trial is succees E PGSSSSF) = PCS) PS) PCS) PCS) PLR) because Each trial is independent of the other CS Scanned with CamScanner