UIT-III SOLVED EXAMPLES AVERAGES- MEASURES OF CETREL TEDECY a. Calculate the median and mode for the sales from the data given below. Sales(in Lacs): <10 10-15 15-0 0-5 5-30 30& above o of Salesmen: 3 1 16 5 10 4 Working Table: Sales c.f.(<) Median has 50% = observations on l.h.s f <10 3 3 ( ch ) 10-15 1 15 And Md l 70 Where. f = 35; 15-0 16 31 0-5 5 56 Hence class of Median is 0-5 & l=0,f=5,c=31, h=5 5-30 10 66 (35 31)5 We put the values to get, Md 0 =0.8 30& above 4 70= 5 Total 70 ( fm f 1) h ow, Mode= l fm f 1 f Where, fm= 5,l = 0, f1 = 16, f =10, h =5 (5 16)5 We put the values to get, Mode= 0 =.36 x5 16 10 b. Calculation of Median and Quartiles Q1, Q &Q3 Daily wages (Rs.):<100 100-15 15-150 150-175 175-00 00& above o of workers: 3 1 16 5 10 4 Working Table Wages(Rs) ` f c.f.(<) <100 1 1 100-15 40 5 15-150 55 107 150-175 10 117 175-00 5 1 00& above 3 15 = Total 15 Q 1 has 5% = observations on l.h.s 4 ( ch ) And, Q 4 15 1 l Where = f 4 4 31.5 We put the values to get, (31.5 1)5 Q1 100 =11.03 40 Similarly, Q3 has 75%= 3 observations on l.h.s. 4 3 ( ch ) Q 4 3 l where, 3 =93.75, l=15,c=5,f=55& h=5 f 4 (93.75 5)5 Q3 15 =143.97 55 c. Calculate the Quartile Deviation (QD) and it s coefficient Data: Reduction in weight: 0-0 0-40 40-60 60-80 80-100 o of patients 1 3 35 0 10 Page 1 of 17
Calculation Table: C.I. f c.f.(<) 0-0 1 1 0-40 3 3 40-60 35 70 60-80 0 90 80-100 10 100= Total 100 Q 1 has 5% = obs on l.h.s. 4 ( ch ) And, Q 4 1 l f 100 Where = = 5, l=0, c=1,f=3,h=0 4 4 (5 1)0 We put the values to get, Q1 0 =31.30 3 3 ( IIIy, Q3 has 75%= 3 ch ) obs on l.h.s. And, Q 3 l 4 4 f Where, 3 (75 70)0 =75, l=60,c=70,f=0& h=0 Q 3 60 =65 4 0 Q3 Q1 65 31.30 Q ow Q.D= = 16.85& C.Q.D. = 3 Q 1 65 31.30 0.35 Q3 Q1 65 31.30 d. CALCULATIO OF S.D. Ex: Compare the performance of SHIKHAR DHAWA/VIRAT KOHLI BASED O LAST 5 ODI Ans: TO COMPARE THE VARIATIOS WE OBTAI CV FOR BOTH, ODI O DHAWA KOHLI SD X VK X ODI # 116 14 3397 116xM+=13456 14xM+=196 ODI # 17 115 3395 89 135 ODI # 16 3388 56 4 ODI # 15 31 3387 5 961 ODI # 69 10 3383 4761 10404 Total 33 64 18987 4790 CLACULATIO FOR DHAWA ΣX 33 Mean= 46.6 ΣX 18987 S.D.= 40.317 CLACULATIOS FOR KOHLI ΣX 64 Mean= 5.8 ΣX 4790 S.D.= 46.585 Page of 17
S.D.= Mean x 33 X = = 5 =46.6 Mean X x 64 = = 5 =5.8 x 18987 X = 46.6 x 4790 =40.3 S.D.= X = 5.8 =46.58 n 5 n 5 C.V.= 86.5 Less C.V.=88. More Hence Dhawan was more consistent than Kohli. S.D. FOR GROUPED DATA: WHE FREQUECIES ARE GIVE, S.D. = fx X Table of calculation:- Total = fx = f DATA: Daily wages: 0-4 5-9 10-14 15-19 0-4 5-9 o of workers: 3 6 16 6 5 4 FOR CV WE FID MEA & SD. WHERE, fx Mean= X = 560 = 40 =14 S.D.= fx X 9680 = 14 40 =6.78 And, C.V.= C.I. f X Fx f x e.g. 10-15 8 10 15 =1.5 SD.. x100 =48.44 Mean 8x1.5 =100 100x1.5 = 150 ot 100x100 DATA CALCULATIO OF S.D. Daily wagesin100rs. o of workers: f X Mid point fx fx 0-4 3 3X=6 6x =1 5-9 6 7 4 94 10-14 16 1 19 304 15-19 6 17 10 1734 0-4 5 110 40 5-9 4 7 4X7=108 916 Total =40 Σfx=560 Σfx =9680 x = Page 3 of 17
Ex:. The data given below are the score of students in a common examination test (CET). Calculate the Standard Deviation (S.D.). Also find the coefficient of variation. Score: 0-50 50-100 100-150 150-00 00-50 50-300 o of Student: 6 15 75 45 14 5 Ans: Table of s.d. calculation Score f X Fx f 0-50 50-100 100-150 150-00 00-50 50-300 6 15 75 45 14 5 Total = 70 5 75 15 175 5 75 150 9375 9375 7875315 01375 fx= 31300 x 3750 70315 1171875 137815 708750 37815 f x fx Where, Mean X = = 31300 70 =115.9, S.D = fx X 4343750 = 115.9 51.48 70 SD.. C.V. = x100 = 51.48 x100= 44.4 Mean 115.9 Ex:. Calculate the standard deviation (S.D.) from the data given below. Monthly rent paid (000Rs): 0-5 5-10 10-15 15-0 0-5 o of families: 4 3 8 15 C.I f x Fx fx 0 to 5 4.5 10 5 fx Where, Mean X = = 830 5 to 10 3 7.5 17.5 193.75 7 =11.5, 10 to 15 15 to 0 0 to 5 8 1.5 15 17.5 4343750 350 4375 6.5 4593.75.5 45 101.5 7= 840 fx 11300 fx S.D = =4.56 fx X = 11300 11.66 7 Page 4 of 17
UIT-IV PROBABILITY a. Coin problem: A group of 3 coins is tossed as a time, find the probability of getting, i. Exactly Heads ii. At most Heads Solution: When three coins are tossed up at a time the sample is S = HHH, HHT, HTH, THH,HTT,THT, TTH, TTT n(s) =8. Event A: Exactly two Head turns up. A= { HHT, THH, HTH} n(a)= 3.. na. P(A) = ( ) ns ( ) = 3 8 Event B: At most two Heads:.B: HHT, HTH, THH,HTT,THT, TTH, TTT n(b) =7... nb ( ). P(B) = ns ( ) = 7 8. b. Dice Problem: A pair of fair dice is rolled. Write down the sample space and find the probability that, the sum of dots on the uppermost face is i) 6 or 10. ii) Multiple of 4. iii) < 6. Solution: When a pair of dice is rolled, the sample space is S= (1,1), (1,), (1,3),( 1,4), (1,5), (1,6), (,1) (,6).. (5,6), (6,6) n(s)= 36. To find the probability we define the events, i) Event A: the sum of dots on the uppermost face is 6 or 10. A= {(1,5) (5,1), (,4) (4,) (3,3) (4,6) (6,4) (5,5)} n(a)= 8... na. P(A) = ( ) ns ( ) = 8 36 ii) event B: The sum of the dots on the uppermost faces is divisible by 4. B: {(1,3),(3,1),(,),(,6),(6,),(3,5),(5,3),(4,4),(6,6)} n(b)=9... nb ( ). P(B) = ns ( ) = 9 36 =0.5. iii) event C: the sum of the dots is < 6. C: { (1,1),(1,),(1,3),(1,4),(4,1)(,1),(,),(,3),(3,1),(3,)} n(c) = 10... nc ( ). P(C)= ns ( ) = 8 36. UIT-V: DECISIO THEORY a. Solve the Decision problem using MAXIMAX, MAXIMI & LAPLACE criteria Pay-off Table MAXIMAX MAXIMI MAX(AVERAGE) Stock/Demand 90 100 110 10 Col A Max Col B Min Col C Average A1-90 50 50 50 50 MAX 50 50 TOATL/4= 50 A- 100 1000 500 500 500 500 1000 15 A3-110 -50 150 750 750 750-50 165 A4-10 -1500 0 1500 3000 MAX 3000-1500 750 Total 3000-1500 50 Page 5 of 17
b. Solve the Decision problem using MAXIMAX, MAXIMI & LAPLACE criteria Action/ States of nature A1 A A3 Optimal Decision S1 50 10 100 S 700 500 60 S3 500 900 80 Max 700 MAX 900 100 A Min 50 10 MAX 60 A3 Average 416.667 MAX 470 80 A c. Solve the Decision problem using EMV criteria EMV criteria: We calculate the EMV values as follows EMV (A!) = 0x0.3 +5x0.4 +(-1)x0.3= 7.7---Max EMV (A) = 8x0.3 +5x0.4 + 4x0.3 = 5.6 EMV (A3) = -10x0.3 +5x0.4 +10x0.3= Hence the optimum decision is A1 Probability 0.3 0.4 0.3 d. Solve the decision problem using Minimax Regret criteria Pay-off Table Regret Table Mark max for the States of nature Action/ States of A1 A A3 A1 A A3 nature Action Pay-off Table State of nature S1 S S3 A1 0 5-1 A 8 5 4 A3-10 5 10 S1 50 10 100 100-50=50 90 0 S 700 500 60 700-700=0 00 630 S3 500 900 80 900-500=400 0 810 Max regret 400 00 Min(Max) 810 e. Solve the Decision problem using EOL criteria Pay-off We prepare the Regret (OL) table as follows Action State of nature S1 S S3 A1 0 0 11 A 8 0 6 A3-10 0 0 Action State of nature S1 S S3 A1 0-0 = 0 0 11-11=0 A 0-8= 1 0 11-6=5 A3 0-(-10)=30 0 11-0-11 EOL (A1) = 0x0.3+0x0.4+11x0.3 =3.3----Min EOL EOL (A) = 1x0.3+0x0.4+6x0.3 =5.4 EOL (A3) = 30x0.3+0x0.4+11x0.3 = 1.3 Optimal Decision: A1 Page 6 of 17
TUTORIAL ASSIGMET:-I Unit-II: PERMUTATIOS & COMBIATIOS Q1 Evaluate the following, i) 5 P 3 + 7 P ii) 8 P 3 + 6 P 4 iii) 10 P 8 + 10 P 7 iv) 8 C 4 + 7 C 3 v) 8 C 6 + 8 C 7 Q. In how many possible the letters in the word FATHER be arranged so that, a) All the vowels are always together b) they are not together Q3. Five books on Mathematics, 4 books on English & 3 books on History are to be put in a shelf in a row. In how many possible ways can this be done so that, Books of same subjects are always together Only English books are together o two Mathematics are together Q4. A box contains 6 Green & 5 Red balls, a pair of balls is drawn at random. Find the no of possible selections so that, Both the balls are of same colours. They are of different colours Only red balls are drawn Q5. In how many possible ways 3 cards can be drawn from the pack of 5 cards so that, i) all 3 are Ace cards; ii) there are two kings and one queen iii) cards are of same suit TUTORIAL ASSIGMET:-II Unit-IV: PROBABILITY & RADOM VARIABLES 1. A cubic die is rolled down. What is the probability of getting, a) o of dots <4 b) no of dots as multiple of 3. A group of 3 coins is tossed up at a time. Find the probability that, a) Only 1H turns up b) there are more H than T 3. A pair of unbiased dice is rolled down. Find the probability that, a) sum of the dots is <6 b) the sum of the dots is 7 or 11 4. 3 cards are drawn from the pack of 5 cards. Find the probability that, a) all 3 are Ace cards b) all are of same suit c) there are kings & 1 queen 5. Given P(A)= 0.5, P(B) = 0.6 & P (A B) = 0.4 Find, i) P(A B) ii) P(A/B) iii) (only A) 6. For independent events A & B, P(A) = ½, P(B) = ¾. Find, i) P(A B) ii) P(only B) &iii) (only A) iv) P(Only One) 7. A problem on Maths is given to students A, B who attempt it independently. What is the probability that, i) the problem is solved? ii) it is solved by only one? Given that their chances of solving are 1/3, & 3/4 respectively. Page 7 of 17
TUTORIAL O:III Unit-IV:EXPECTED VALUE & VARIACE 1. Find the expected value & variance of the r.v. X defined as The no of Heads in the experiment of tossing a unbiased coin four times.. Find the Mean & Variance of the r.v. X from the following probability distribution. X 5 10 15 0 5 p(x) 0.1 0.3 0.35 0. 0.1 3. Find the value of k so that the given p(x) represents a probability distribution. Hence find the expected value of X. X: 4 5 6 7 8 p(x): 0.15 0.0 k 0.15 0.10 4. Find the Expected value and Variance of the r.v. X for the given probability distribution. Hence find the expected value of X. X: -1 0 1 3 4 P(x): 0.10 0.05 0.15 0.15 0.30 0.5 5. A man tosses a cubic die in a fun & fair game. According to the terms of the game. He earns, Rs10/- if the of the dots is multiple of 3, Rs15/- if the no of dots is less than 3 & earns nothing otherwise. Find his expected gain from the game if he has to pay Rs10/- as the entry fee. TUTORIAL O:IV Unit-V:DECISIO THEORY-I 1. Suppose that a decision maker faced with three decision alternatives (Acts) and three state of nature(events) with the following pay-off table: Action State of nature E1 E E3 A1 15 0 6 A 18 10 15 A3 5 16 9 Solve the Decision problem using, a) Maximin b) Minimax regret c) Minimax Page 8 of 17
. Solve decision problem using Minimax regret criterion: Event E1 E E3 Action A1 5 10 18 A 8 8 A3 1 18 1 A4 30 7 19 3 Determine the best decision according to EMV criterion. Action/Events E1 E E3 A1 10 1 5 A 18 4 10 A3 5 30 0 Probability: 0. 0.5 0.3 TUTORIAL O:V Unit-V:DECISIO THEORY-II 1. Draw a decision tree diagram to show the solution to the decision problem using EMV criterion: Event A1 A P(E) Action E1 5 10 5% E 8 0% E3 1 18 15% E4 30 7 40%. Draw a decision tree for the decision problem below and state the best possible decision. Use EMV criteria. Product/Market Poor Average Good demand P 100 350 100 Q 150 50 150 P(Demand) 30% 55% 15% Page 9 of 17
TUTORIAL ASSIGMET:-VI Unit-III: AVERAGES (MEASURES OF CETRAL TEDECY) 1. The score of students in a class test are given below. Find the values of Mean, Median & Mode. Also count the no of students with score above the Mean score. 18 0 1 15 15 15 16 8 9 11 11 6 0 14 15 13 18 19 10 14. The closing price of shares on 5 trading days of the market are as follows, Calculate the mean price. Closing price(rs): 30-34 35-39 40-44 45-49 50-54 o of shares: 18 35 45 3 5 3. The Height of students in a class is given below. Calculate the values of Mean & Median. Height in cms: 110-10 10-130 130-140 140-150 150-160 o of students: 11 35 45 14 5 4. Calculate the combined mean from the data given below. Sample I II o of items: 100 150 Means weight(kgs): 30 3 5. The daily wages paid to the workers are given below. Wages in Rs: <50 50-300300-350 350-400 400-450 450& above O of Workers: 10 30 48 15 5 Calculate the Median & 3 Quartiles. Hence state the wage limit that covers middle 50% of workers. TUTORIAL ASSIGMET:-VII Unit-III: DISPERSIO(MEASURES OF VARIATIO) 1. Calculate the QD &it s coefficient. Sales Range (in LacsRs.) : Below35 35-40 40-45 45-50 50+ o of salesmen: 8 43 55 3 1. The score of candidates in a CAT examination is shown below. Calculate S.D. & C.V. Score: 75-100 100-15 15-150 150-175 175-00 o of Candidates: 9 1 35 18 1. The data given below read the price range of car sales over the period of six months. Calculate the Coefficient of Variation. Price (in Lacs Rs.):.5-7.5 7.5-1.5 1.5-17.5 17.5-.5.5-7.5 o of cars sold (in 100): 3 17 33 16 6 3. Calculate the combined S.D. for the data below Sample I II o of items: 100 150 Mean weight (kgs): 15 16 S.D. 3 4 Page 10 of 17
TUTORIAL O:X Unit-II:LIEAR PROGRAMMIG PROBLEM 1. Solve the following LPP by graphical method. Min Z= 150x+100y s.t. 6x+y 6; x+4y 6; x & y 0. Solve the following LPP by graphical method. Max Z= 6x+5y s.t. 4x+5y 0; x+6y 1; x& y 0. 3. Solve the following LPP by graphical method. Min Z= 10x+15y; s.t. 3x+y 3; x+y 3; x+y 4; x& y 0 4. Solve the following LPP by graphical method. Max Z= 16x+15y s.t. 4x+5y 0; x+6y 1; x& y 0 TUTORIAL O: VII Unit-I: SAHRES & MUTUAL FUDS 1. Calculate the brokerage @ 1.5% on the purchase of 300 shares @ Rs.65/-.. Calculate the dividend earned @40% on 50 shares of FV 5/- purchased @ 90/- 3. If a dividend of Rs. 150 is earned on 150 shares of FV /-. Find the rate of dividend earned. 4. Mr. Akash purchased 00 shares of HDFC @ 575/- & sold for 650/- each after receiving a dividend of 40% on FV 5/-. Calculate the % profit to him if he paid % brokerage. 5. Miss. Babita sold 00 Rs 10/- shares @ Rs.90/-.She invested the amount in buying 150 other shares @15/- Find the extra amount required if any, when the brokerage paid was 1.5% 6. Miss. Anju invested Rs.54000 in buying certain ten rupee shares @ Rs.90/-. He sold 1/3 rd of them @ 15/- after 10 days and the rest @ 110/- at the end of year after receiving a dividend of 10%. Find the gain to her in the transaction. TUTORIAL O: IX Unit-I: SAHRES & MUTUAL FUDS 1. Calculate the amount of entry load applicable @.5% on the purchase of 500 units at AV Rs. 48.55/-.. Mr Akash sold 400 units of HDFC mutual fund at AV Rs. 15/- & purchased SBI mutual fund at AV Rs.78.5/-. Find the no of units purchased when load of.5% was applied in both the transactions. 3. If the AV of a MF unit is increased from 8 to 45 in one year, what is the % growth? 4. Anand invested Rs.1Lac in the gift fund of HDFC Mutual fund at AV of Rs. 8/-.He sold the units at AV Rs.3/-after receiving a dividend @ 5%. Find the net % gain to him, if the load was.5% on both the transaction Page 11 of 17
QUESTIO BAK SEM-I SECTIO-I (MATHS) Unit-I: SHARES & MUTUAL FUDS 1. Mr. Ajay invested Rs.45000 in buying certain ten rupee shares @ Rs.90/-.He sold half of them @ 100/- after 10 days and the rest @ 80/- at the end of year after receiving a dividend of 10%. Find the gain or loss to Mr. Ajay in the transaction.. Calculate the sale value of 500 shares @ Rs. 45/- if brokerage 0.3%.is applied. 3. Calculate the amount of brokerage paid on the purchase of 50 shares of FV Rs.10/-@ Rs.75/- when brokerage charged is 0.35%. Also calculate the purchase value. 4. Calculate the amount of dividend @ 40% earned on 00 shares of F.V. Rs 5/- which were purchased at Rs.55/- each. 5. Mr. Rajesh bought 500 shares of FV 5/- @ Rs.150/-. He sold 40% of them @ 180/- and the rest @ 5/- at the end of year after receiving a dividend of 5%. Find the net % gain to Mr. Rajesh in the transaction. Brokerage of 1.5 % is applied on both the transaction. 6. Mr. Shah invested Rs.43680/- in buying certain ten rupee shares @ Rs.91/-. He sold half of them @ 100/- and the rest @ 80/- at the end of year after receiving a dividend of 10%. Find the gain or loss to Mr. Shah in the transaction. 7. The AV of a mutual fund increased from 48 to 64 within a year. Calculate the rate of growth. 8. Calculate the amount entry load @1.5%.applied on the purchase of 00 units of a Mutual Fund at AV Rs.75/- 9. A person invest Rs.1,00,000 in the gift fund of HDFC Mutual fund on 11//007. Find the no of units purchased by him at AV Rs. 15/- with entry load of.5%. 10. Suppose a scheme with 1,000 units ha the following items in its balance sheet: Unit Capital Rs. 10,000; Investments at market value Rs. 5,000; Other assets Rs. 3,500; Other liabilities Rs.,000; Issue expenses not written off Rs. 500; Reserves Rs. 17,000. What would be its AV? Unit-II: PERMUTATIOS & COMBIATIOS 1. Evaluate the following, 5 P 3 + 7 P ii) 8 P 3 + 6 P 4 iii) 10 P 8 + 10 P 7. In how many possible the letters in the word ATTITUDE be arranged so that, All the vowels are always together 3. All the consonants are togetherfour books on PHYSICS 3 books on CHEMISTRY & books on BIOLOGY are to be put in a shelf in a row. In how many possible ways can this be done so that, Books of same subjects are always together Only BIOLOGY books are together BIOLOGY books are at end position Page 1 of 17
4. Six boys & Girls are to stand in a row for a group photo. How possible ways they can have a photo so that, Girls always stand together They do not stand together They stand at the end position? 5. From the digits 1,,5,6,8 & 9 a 3 digits number is to be formed. In how many possible this can be done so that, o digit is repeated Digits are allowed to repeat Only even number without repeated is formed. 6. In how many possible ways balls can be drawn out of 15 balls? 7. Find the no of possible ways to draw a pair of cards from the pack of 5 playing cards. 8. A box contains 6 Green & 5 Red balls, a pair of balls is drawn at random. Find the no of possible selections so that, Both the balls are of same colours. They are of different colours 9. Out of 6 Batsmen, 5 Bowlers & 3 Wicketkeeper a Team of 11 players is to be formed. How many ways can this be done so as to include, 5 Batsmen, 4 Bowlers & Wicketkeeper 6 Batsmen, & at most 1 Wicketkeeper 10. In how many possible ways 3 cards can be drawn from the pack of 5 cards so that, i) all 3 are Ace cards; ii) there are two kings and one queen. iii) cards are of same suit. 11. Solve the following LPP by graphical method. i) Max Z= 6x+7y s.t. 4x+5y 0; x+6y 1; x & y 0. ii) Min Z= 10x+15y s.t. 3x+y 3; x+y 3; x & y 0 iii) Max Z= 0x+30y s.t. 3x+3y 36; 5x+y 50; x+y 3. iv) Max Z= 0x+30y s.t. 4x+6y 4; 3x+7y 1; x+y 5; x & y 0. v) Min Z= 1+0y s.t. x+y 3; x+y 3; x+y 4; x & y 0 vi) Min Z= 10x+15y s.t. 3x+y 3; x+y 3; x & y 0 Page 13 of 17
SECTIO-II (STATS) Unit-III: AVERAGE & DISPERSIO 1. What are the positional averages. How can they be approximated?. Write short note on Partition values. 3. Calculate the values of Mean, Median and Mode. Height in cms: 10-0 0-30 30-40 40-50 50-60 o of students: 18 35 45 1 5 4. The sum of deviations of the x values in a certain group of observations taken from 5 is 10 and that from 35 is -880. Find the mean and no of observations in the group. 5. Calculate Median and Mode for the following distribution. Income in Rs.: 5-10 10-15 15-0 0-5 5-30 o of Person: 30 50 100 40 30 6. Calculate 3 Quartiles from and hence state the salary limits that include middle 50% of the employees. Salary(per day Rs): Below 100 100-00 00-300 300-400 400-500 500-600 o of Employees: 5 1 3 35 0 5 7. Calculate the Q.D. & it s coefficient from the data given below. Wages : <800 800-1000 1000-100 100-1400 1400-1600 o of Workers: 0 60 80 30 10 8. Calculate the Q.D. & it s coefficient from the data given below. Age : <5 5-9 30-34 35-39 40-44 45& above o of Workers: 10 30 48 15 5 9. The coefficient of Q.D. for a certain group of observations is 0. and the sum of lower and upper quartiles is 100. Fine the two quartiles and the Q.D. 10. Find the mean deviation and it s coefficient from mode for the data given below. Price of shares: 00-0 0-40 40-60 60-80 80-300 o of shares: 4 3 50 17 7 11. Calculate the standard deviation (S.D.) and C.V. from the data given below. Increase in height (in cms): 0- -4 4-6 6-8 8-10 o of children 1 3 35 0 10 1. Calculate the coefficient of variation for the data given below. Earning per share(in 100 Rs): 0-5 5-10 10-15 15-0 0-5 o of Shares: 5 1 0 10 8 13. Find the coefficient of variation for the data given below. Daily wages: 00-0 0-40 40-60 60-80 80-300 o of workers: 4 3 50 17 7 Page 14 of 17
14. Compare the two groups on consistency level & state which group is more consistent. Group A B Mean 75 40 S.D. 8 5 Also calculate the combined S.D. 15. Calculate the combined standard deviation (S.D.) from the data given below. Sample I II o of items: 100 150 Means: 15 16 S.D. 3 4 Unit-IV: PROBABILITY 16. Find the probability of getting a prime number when a cubic die is tossed up. 17. A box contains 0 tickets numbered 1-0. A ticket is drawn at random form the box. Find the probability that the number on the ticket is, a perfect square a multiple of 3 18. A pair of unbiased dice is rolled at a time find the probability that the sum of the dots appearing on the uppermost faces is, i) 6 or 10; ii) multiple of 4; iii) 10 or more. 19. Three unbiased coins are tossed up at a time. Find the probability that, i) exactly Heads appear; ii) at most Heads appear. 0. Three cards are drawn from the pack of 5 cards. Find the probability that, i) all 3 are Ace cards; ii) there are two kings and one queen; iii) all are face cards iv) cards are of same color 1. Find the probability of getting a Face card when a card is drawn at random from a pack of 5 playing cards.. Given P (A) = /5, P (B) = 3/4 & P (A B) = ½. Find, i) P (AUB); ii) P (A/B); iii) P (B/A) Are events A & B independent? 3. Given P(A)= 0.5, P(B)= 0.7 & P(A B) = 0.4 Find, i) P (AUB); ii) P (A/B) iii) P( only A) iv) P(only one) 4. Find the expected value & variance of X from the probability distribution given below X: 1 3 4 5 p(x): 0.15 0.0 0.4 0.15 0.10 UIT V:-DECISIO THEORY 5. Define the terms for a Decision Problem i) Course of Action ii) State of ature iii) Pay-off Page 15 of 17
6. Explain the terms: Maximax criteria Laplace criteria Maximin criteria 7. What is the Regret or Opportunity loss? How is it used to find optimum decision? 8. Solve the given decision problem using, a) Maximin b) Minimax c) Laplace criteria. Course of Action States of nature(events) S1 S S3 A 50 10 5 B 180 40 100 C 15 30 00 9. Obtain the best decision using Minimax criteria Action Events E1 E E3 A1 10 1 5 A 18 4 10 A3 5 30 0 30. Solve the Decision problem using i) Minimax ii) Maximax & iii)laplace criteria EVETS COURSE OF ACTIO A1 A A3 E1 0 5 15 E 8 5 4 E3 10 5 10 31. Given the pay-off matrix, solve the decision problem using, i) Laplace ii) Maximin iii) Maximax States of nature/ Action A1 A A3 S1 50 700 500 S 10 500 900 S3 100 60 80 3. Given the pay-off matrix, solve the decision problem using, i) Laplace ii) Maximin iii) Maximax Action/Event E1 E E3 A1 10 5 16 A 5 1 18 A3 30 15 A4 15 5 0 Page 16 of 17
33. Determine the best decision according to EMV criterion. States of nature Course of action A1 A A3 E1 10 1 5 E 18 4 10 E3 5 30 0 Given, P(E1)= 0.4 P(E)= 0.5 P(E3)= 0.1 34. Draw a decision tree for the decision problem below and state the best possible decision. Action Events E1 E E3 A1 15 8 16 A 1 0 15 Probability: 0. 0.5 0.3 35. Draw decision tree for the following problem and suggest a best course of action (Use EMV) Action Choice of product States of Demand & profit Fair Good Best P 900 90 500 Q 100 500 700 Probability 0. 0.5 0.3 DO TRY ALL THE PROBLEMS O YOUR OW. Page 17 of 17