Sales c.f.(<) Median has 50% = observations on l.h.s. (35 31)5 = & above 4 70=N. We put the values to get,

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1 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): < & above o of Salesmen: Working Table: Sales c.f.(<) Median has 50% = observations on l.h.s f < ( ch ) And Md l 70 Where. f = 35; Hence class of Median is 0-5 & l=0,f=5,c=31, h= (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 x b. Calculation of Median and Quartiles Q1, Q &Q3 Daily wages (Rs.):< & above o of workers: Working Table Wages(Rs) ` f c.f.(<) < & above 3 15 = Total 15 Q 1 has 5% = observations on l.h.s 4 ( ch ) And, Q l Where = f We put the values to get, (31.5 1)5 Q1 100 = 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 ( )5 Q3 15 = c. Calculate the Quartile Deviation (QD) and it s coefficient Data: Reduction in weight: o of patients Page 1 of 17

2 Calculation Table: C.I. f c.f.(<) = 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 = ( 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 = Q3 Q Q ow Q.D= = 16.85& C.Q.D. = 3 Q Q3 Q 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 # xM+= xM+=196 ODI # ODI # ODI # ODI # Total CLACULATIO FOR DHAWA ΣX 33 Mean= 46.6 ΣX S.D.= CLACULATIOS FOR KOHLI ΣX 64 Mean= 5.8 ΣX 4790 S.D.= Page of 17

3 S.D.= Mean x 33 X = = 5 =46.6 Mean X x 64 = = 5 =5.8 x 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: o of workers: FOR CV WE FID MEA & SD. WHERE, fx Mean= X = 560 = 40 =14 S.D.= fx X 9680 = =6.78 And, C.V.= C.I. f X Fx f x e.g =1.5 SD.. x100 =48.44 Mean 8x1.5 = x1.5 = 150 ot 100x100 DATA CALCULATIO OF S.D. Daily wagesin100rs. o of workers: f X Mid point fx fx X=6 6x = X7= Total =40 Σfx=560 Σfx =9680 x = Page 3 of 17

4 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: o of Student: Ans: Table of s.d. calculation Score f X Fx f Total = fx= x f x fx Where, Mean X = = =115.9, S.D = fx X = SD.. C.V. = x100 = x100= 44.4 Mean Ex:. Calculate the standard deviation (S.D.) from the data given below. Monthly rent paid (000Rs): o of families: C.I f x Fx fx 0 to fx Where, Mean X = = to =11.5, 10 to to 0 0 to = 840 fx fx S.D = =4.56 fx X = Page 4 of 17

5 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) = nc ( ). P(C)= ns ( ) = UIT-V: DECISIO THEORY a. Solve the Decision problem using MAXIMAX, MAXIMI & LAPLACE criteria Pay-off Table MAXIMAX MAXIMI MAX(AVERAGE) Stock/Demand Col A Max Col B Min Col C Average A MAX TOATL/4= 50 A A A MAX Total Page 5 of 17

6 b. Solve the Decision problem using MAXIMAX, MAXIMI & LAPLACE criteria Action/ States of nature A1 A A3 Optimal Decision S S S Max 700 MAX A Min MAX 60 A3 Average MAX 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= Max EMV (A) = 8x0.3 +5x x0.3 = 5.6 EMV (A3) = -10x0.3 +5x x0.3= Hence the optimum decision is A1 Probability 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 A A A S = S = S = Max regret 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 A A A Action State of nature S1 S S3 A1 0-0 = =0 A 0-8= =5 A3 0-(-10)= EOL (A1) = 0x0.3+0x0.4+11x0.3 = 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

7 TUTORIAL ASSIGMET:-I Unit-II: PERMUTATIOS & COMBIATIOS Q1 Evaluate the following, i) 5 P P ii) 8 P P 4 iii) 10 P P 7 iv) 8 C C 3 v) 8 C 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 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

8 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 p(x) Find the value of k so that the given p(x) represents a probability distribution. Hence find the expected value of X. X: p(x): k Find the Expected value and Variance of the r.v. X for the given probability distribution. Hence find the expected value of X. X: P(x): 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 A A A Solve the Decision problem using, a) Maximin b) Minimax regret c) Minimax Page 8 of 17

9 . Solve decision problem using Minimax regret criterion: Event E1 E E3 Action A A 8 8 A A Determine the best decision according to EMV criterion. Action/Events E1 E E3 A A A Probability: 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 E % E 8 0% E % E %. 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 Q P(Demand) 30% 55% 15% Page 9 of 17

10 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 The closing price of shares on 5 trading days of the market are as follows, Calculate the mean price. Closing price(rs): o of shares: The Height of students in a class is given below. Calculate the values of Mean & Median. Height in cms: o of students: Calculate the combined mean from the data given below. Sample I II o of items: Means weight(kgs): The daily wages paid to the workers are given below. Wages in Rs: < & above O of Workers: 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.) : Below o of salesmen: The score of candidates in a CAT examination is shown below. Calculate S.D. & C.V. Score: o of Candidates: 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.): o of cars sold (in 100): Calculate the combined S.D. for the data below Sample I II o of items: Mean weight (kgs): S.D. 3 4 Page 10 of 17

11 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 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 1.5% on the purchase of 300 Rs.65/-.. Calculate the dividend on 50 shares of FV 5/- 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 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/- Rs.90/-.She invested the amount in buying 150 other Find the extra amount required if any, when the brokerage paid was 1.5% 6. Miss. Anju invested Rs in buying certain ten rupee Rs.90/-. He sold 1/3 rd of 15/- after 10 days and the 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 on the purchase of 500 units at AV Rs /-.. 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 5%. Find the net % gain to him, if the load was.5% on both the transaction Page 11 of 17

12 QUESTIO BAK SEM-I SECTIO-I (MATHS) Unit-I: SHARES & MUTUAL FUDS 1. Mr. Ajay invested Rs in buying certain ten rupee Rs.90/-.He sold half of 100/- after 10 days and the 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 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 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 Rs.150/-. He sold 40% of 180/- and the 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 Rs.91/-. He sold half of 100/- and the 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 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 P ii) 8 P P 4 iii) 10 P 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

13 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

14 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: o of students: The sum of deviations of the x values in a certain group of observations taken from 5 is 10 and that from 35 is Find the mean and no of observations in the group. 5. Calculate Median and Mode for the following distribution. Income in Rs.: o of Person: Calculate 3 Quartiles from and hence state the salary limits that include middle 50% of the employees. Salary(per day Rs): Below o of Employees: Calculate the Q.D. & it s coefficient from the data given below. Wages : < o of Workers: Calculate the Q.D. & it s coefficient from the data given below. Age : < & above o of Workers: 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: o of shares: Calculate the standard deviation (S.D.) and C.V. from the data given below. Increase in height (in cms): o of children Calculate the coefficient of variation for the data given below. Earning per share(in 100 Rs): o of Shares: Find the coefficient of variation for the data given below. Daily wages: o of workers: Page 14 of 17

15 14. Compare the two groups on consistency level & state which group is more consistent. Group A B Mean 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: Means: 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 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: p(x): 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

16 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 B C Obtain the best decision using Minimax criteria Action Events E1 E E3 A A A Solve the Decision problem using i) Minimax ii) Maximax & iii)laplace criteria EVETS COURSE OF ACTIO A1 A A3 E E E Given the pay-off matrix, solve the decision problem using, i) Laplace ii) Maximin iii) Maximax States of nature/ Action A1 A A3 S S S Given the pay-off matrix, solve the decision problem using, i) Laplace ii) Maximin iii) Maximax Action/Event E1 E E3 A A A A Page 16 of 17

17 33. Determine the best decision according to EMV criterion. States of nature Course of action A1 A A3 E E E Given, P(E1)= 0.4 P(E)= 0.5 P(E3)= Draw a decision tree for the decision problem below and state the best possible decision. Action Events E1 E E3 A A Probability: 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 Q Probability DO TRY ALL THE PROBLEMS O YOUR OW. Page 17 of 17

Decision Making. DKSharma

Decision Making. DKSharma Decision Making DKSharma Decision making Learning Objectives: To make the students understand the concepts of Decision making Decision making environment; Decision making under certainty; Decision making