Probability – Solved Worksheets for Semester 2 (Commerce), Study notes of Business Mathematics

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Tabulation of data:

1. In a sample survey study about the drinking habits in two cities, it is observed that, in

city X 57% are male, 22% are drinkers, and 14% are male drinkers, whereas in city Y

52% are male, 28% are drinkers and 21% are male drinkers. Tabulate the above

information.

2. Present the following information in a suitable tabular form: In 2009 out of a total

2000 employees in a company 1550 were members of a trade union. The number of

women employees was 250, out of which 200 did not belonging to any trade union.

While in 2010 the number of union employees was 1725 out of which 1600 were

men. The number of none union employees was 380 among which 155 were women.

3. Out of total number of 5000 people who applied for admission of their child in a

school 3950 belongs to general category and remaining belong to the EWS category.

in EWS category 275 parents belong to service class and remaining were self-

employed. The number of self employed applicants belonging to general category are

725. Tabulate the given information.

4. In 2005 out of a total of 1750 workers of a factory 1200 workers were members of a

trade union. The number of women employed was 200 of which 175 did not belong

to trade union. In 2010, the number of union workers increased to 1580 of which

1290 were men. On the other hand, the number of non-union workers fell down to

208 of which 180 were men. In 2015, there were on the pay rolls of factory, 1850

workers of who 1800 belong to a trade union. Of all the employees in 2015, 300 were

women of whom only 8 did not belong to a trade union.

6. (Question+ answer)

Trip Days Expenses (₹. ’00) Expenses per day (₹. ’00) 1 0.5 13.50 27 2 2 12.00 6 3 3.5 17.50 5 4 1 9.00 9 5 9 27.00 3 6 0.5 9.00 18 7 8.5 17.00 2 Total 25 105.00 70 An auditor criticized these expenses as excessive, asserting that the average expenses per day is ₹.

1000. The sales intern replied that the average is only ₹. 420 and that in any event the median is the appropriate measure and is only ₹. 300. The auditor rejoined that the arithmetic mean is the appropriate measure but that the median is ₹. 600. You are required to i. Explain the proper interpretation of each of the averages mentioned. ii. Which average seems appropriate to you?
1001. Calculate the median from the following data: Marks (more than) 70 60 50 40 30 20 No. of students 7 18 30 50 63 70
1002. The distribution of processing time of jobs on a computer is given below. Compute Q1, D4, D9, P12, P Processing time (in secs) 15 - 20 20 - 25 25 - 30 30 - 35 35 - 40 40 - 45 45 - 50 Frequency 3 11 29 53 23 14 6
1003. Calculate range and its coefficient from the following data: 53, 46, 18, 16, 75, 84, 28
1004. Following are the marks obtained by two students A and B in 10 tests of 100 marks each: A 44 80 76 48 52 75 75 51 60 54 B 48 75 54 60 63 69 72 51 57 66 Find out who is better in studies and if consistency is the criteria for awarding a prize who should get the prize?
1005. Compute quartile deviation from the following marks of 12 students: 37,78,86,91,93,94,35,42,53,55,57.
1006. The following table gives the age distribution of boys and girls in a school. Find which of the two groups is more variable in age: Age: 10 11 12 13 14 15 No. of boys: 11 14 14 10 8 5 No. of girls: 13 15 12 9 5 3
1007. Find Mean deviation from median and coefficient of mean deviation from the following data: Age: 18 20 22 24 26 28 30 No. of persons:12 15 20 25 18 10 8
1008. Calculate Quartile deviation: Value 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75- 85 85 - 95 Frequency:32 38 45 98 122 80 50 25
1009. Following information relates to two batsman A and B. Batsman A Batsman B No. of innings played 10 10

Average runs scored per innings 45.5 52. Standard deviation 10 11 Find out a) who is better run getter? b) who is consistent batsman? Use coefficient of variation.

21. Following data represents lives of 2 models of refrigerators A and B. Life(in years) 0 - 2 2 - 4 4 - 6 6 - 8 8 - 10 10 - 12 Model A: 5 16 13 7 5 4 Model B: 2 7 12 19 9 1 i. Find the average life of each model ii. Which model has great uniformity of life?
22. Calculate standard deviation from the following data: Class: 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 Frequency 3 5 15 10 3 4
23. The number of cheques cashed each day at the five branches of a bank during the past month has the following frequency distribution: No. of Cheques: 0 - 199 200 - 399 400 - 599 600 - 799 800 - 999 Frequency: 10 13 17 42 18 The general manager, operations, for the bank knows that a standard deviation in cheque cashing of more than 200 cheques per day creates staffing problem at the branches because of the uneven workload. Should the manager worry about staffing next month?
24. Mean and standard deviation of 150 items were 31.6 and 13.65. while checking it was found that an item 52 was wrongly taken as 25. Find the correct mean and standard deviation.
25. Calculate standard deviation for the following data C.I - 4 - - 3 - 3 - - 2 - 2 - - 1 - 1 – 0 0 – 1 1 - 2 2 - 3 3 - 4 4 - 5 F 4 16 30 40 62 121 98 48 16
26. Samples of paper bags from two manufactures A and B are tested by a prospective buyer for bursting pressure and the results are as follows Bursting Pressure 5 - 9.9 10 - 14.9 15 - 19.9 20 - 24.9 25 - 29.9 30 - 34. (in lbs) No. of bags A 2 9 29 54 11 5 No. of bags B 9 11 18 32 27 13 i. Which set of bags has more uniform pressure? ii. If prices are same, which manufacturer’s bags would be preferred by the buyer? Why?
27. A quality control intern tested nine samples of designs A and B of certain bearing for a new electric crane. The following data are the number of hours it took for each bearing to fail when the crane motor was run continuously at maximum output with a load on the winch equivalent to 1.9 times the intended capacity. A: 16 16 53 15 31 17 14 30 20 B: 18 27 23 21 22 26 39 17 28 Suggest which design is best and why?
28. A purchasing agent obtained samples of 60 watt bulbs from two companies. He had the samples tested in his own laboratory for the length of life with the following results: Length of life: 1700 - 1900 1900 - 2100 2100 - 2300 2300 - 2500 2500 - 2700 Company A: 10 16 20 8 6 Company B: 3 40 12 3 2 a. Which company’s bulbs do you think are better in terms of average life? b. If prices of both the companies are same, which company’s bulbs would you buy and why?

Hungarian Method Steps (Rule) Step- 1: If number of rows is not equal to number of columns, then add dummy rows or columns with cost 0, to make it a square matrix. Step- 2: a. Identify the minimum element in each row and subtract it from each element of that row. b. Identify the minimum element in each column and subtract it from every element of that column. Step- 3: Make assignment in the opportunity cost table a. Identify rows with exactly one unmarked 0. Make an assignment to this single 0 by make a square ( [0] ) around it and cross off all other 0 in the same column. b. Identify columns with exactly one unmarked 0. Make an assignment to this single 0 by make a square ( [0] ) around it and cross off all other 0 in the same rows. c. If a row and/or column has two or more unmarked 0 and one cannot be chosen by inspection, then choose the cell arbitrarily. d. Continue this process until all 0 in rows/columns are either assigned or cross off( ). Step- 4: (a) If the number of assigned cells = the number of rows, then an optimal assignment is found and In case you have chosen a 0 cell arbitrarily, then there may be an alternate optimal solution exists. (b) If optimal solution is not optimal, then goto Step-5. Step- 5: Draw a set of horizontal and vertical lines to cover all the 0 a. Tick(✓) mark all the rows in which no assigned 0. b. Examine Tick(✓) marked rows, If any 0 cell occurs in that row, then tick( ✓) mark that column. c. Examine Tick(✓) marked columns, If any assigned 0 exists in that columns, then tick( ✓) mark that row. d. Repeat this process until no more rows or columns can be marked. e. Draw a straight line for each unmarked rows and marked columns. f. If the number of lines is equal to the number of rows then the current solution is the optimal, otherwise goto step- 6 Step- 6: Develop the new revised opportunity cost table a. Select the minimum element, say k, from the cells not covered by any line, b. Subtract k from each element not covered by a line. c. Add k to each intersection element of two lines. Step- 7: Repeat steps 3 to 6 until an optimal solution is arrived.

Transportation Problem Obtain an IBFS to the following TP by (i) NWCR (ii)Lowest cost entry method (iii) VOGEL’S approximation method. a. Source\Destination D1 D2 D3 D4 Availability S1 11 13 17 14 250 S2 16 18 14 10 300 S3 21 24 13 10 400 Requirement 200 225 275 250 b. Source\Destination D1 D2 D3 D4 Availability S1 1 2 1 4 30 S2 3 3 2 1 50 S3 4 2 5 9 20 Requirement 20 40 30 10 100 c. National Oil Company has three refineries and 4 depots. Transportation costs per barrel and requirements are given below. Determine IBFS using NWCR, LCEM, VAM. Source\Destination D1 D2 D3 D4 Capacity S1 5 7 13 10 700 S2 8 6 14 13 400 S3 12 10 9 11 800 Requirement 300 600 700 400 d. A company has three plants locations A, B, C which supply to warehouses D, E, F, G and H. Monthly plant capacities are 800,500and 900 units respectively. Monthly warehouses requirements are 400,400,500,400and 800 units respectively. Unit transportation cost (in Rs.) are given below: plants\warehouses D E F G H A 5 8 6 6 3 B 4 7 7 6 6 C 8 4 6 6 4 Find the Initial distribution for the company in order to minimize the total TP cost by North west corner rule and Vogel’s approximation method. e. Source\Destination D1 D2 D3 D4 Availability S1 6 5 8 8 30 S2 5 11 9 7 40 S3 8 9 7 13 50 Requirement 35 28 32 25 f. Source\Destination D1 D2 D3 D4 Availability S1 21 16 25 13 11 S2 17 18 14 23 13 S3 32 27 18 41 19 Requirement 6 10 12 15 g. Source\Destination D1 D2 D3 D4 Availability S1 19 30 50 10 7 S2 70 30 40 60 9 S3 40 8 70 20 18 Requirement 5 8 7 11

Assignment Problem

1. Solve the following assignment problem persons I II III IV V A 5 3 4 7 1 jobs B 2 3 7 6 5 C 4 1 5 2 4 D 6 8 1 2 3 E 4 2 5 7 1
2. Job J1 J2 J3 J M1 7 5 8 4 Men M2 5 6 7 4 M3 8 7 9 8
3. A hospital wants to purchase three different types of medical equipments and five manufacturers have come forward to supply one or all the three machines. However, the hospital’s policy is not to accept more than one machine from any one of the manufactures. The data relating to the price (in lakhs of rupees) quoted by the different manufactures are given below. Determine how best the hospital can purchase three machines. Machines I II III A 30 31 27 Manufacturers B 28 29 26 C 29 30 28 D 28 31 27 E 31 29 26
4. The ABC Construction Company has five crews. The skills of the crews vary because of the composition of the crews. The company has five different projects currently. The times (in days) taken by the different crews to complete different projects is given in the table below. Find the best assignment of the crews to different projects such that the total time to complete all the projects is minimized.

5. In a textile sales emporium, four salesmen A, B C and D are available to four customers W, X, Y and Z. Each salesman can handle any counter. The service (in hours) of each counter when manned by each salesman is given below: Salesmen Counters A B C D W 41 72 39 52 X 22 29 49 65 Y 27 39 60 51 Z 45 50 48 52 How should the salesmen be allocated to appropriate counters so as to minimize the service time? Each salesman must handle only one counter.
6. An Airline that operates 7 days a week has the time table shown below. Crews must have a minimum layover of 5 hours between Flights. Obtain the paring of flights that minimizes layover time always from home assuming the crew flying from Delhi and Jaipur can be based either at Delhi or Jaipur for any given pairing the crew will be based at the city that results in the smaller layover. Flight No Delhi Depart Jaipur Arrival Flight No Jaipur Depart Delhi Arrival 1 2 3 4

7.00 A.M

8.00 A.M

1.30 P.M

6.30 P.M

8.00 A.M

9.00 A.M

2.30 P.M

7.30 P.M

8.00 A.M

8.30 A.M

12 Noon 8.00 P.M

9.00 A.M

9.30 A.M

6.30 P.M

9.0 .M

7. Consider the problem of assigning four sales people to four different sales regions. Find the optimal allocation of sales person to different sales region. The cell entries represent annual sales figures in lakhs of rupees. Sales Region Salesman

Provide the business case of the assignment problem above giving the total sales output

shipment of transistors containing 100 units requires 1 hour of engineering, 10 hours of direct labour and 2 hours of administration service. To produce 100 resistors are required 1 hour, 4 hours and 2 hours of engineering, direct labor and administration time respectively. To produce one shipment of the tubes (100 units) requires 1 hour of engineering, 5 hours of direct labour and 6 hours of administration. There are 100 hours of engineering services available, 600 hours of direct labour and 300 hours of administration. What is the most profitable mix?

8. A company produces three products P1, P2 and P3 from two raw materials A, B and labour L. One unit of produce P1 requires one unit of A, 3 units of B and 2 units of L. A unit of product P2 requires 2 units of A and B each and 3 units of L, while one unit of P3 needs 2 units of A, 6 units of B and 4 units of L. The company has a daily availability of 8 units of A, 12 units of B and 12 units of L. It is further known that the unit contribution margin for the products is Rs. 3, Rs. 2 and Rs. 5 respectively for P1, P2 and P3. Formulate this problem as a linear programming problem, and then solve it to determine the optimum product mix.

9. TOYCO assembles three types of toys –trains, trucks and cars-using three operations.

The daily limits on the available times for the three operations are 430, 460, and 420 minutes, respectively, and the revenues per unit of toy train, truck and car are Rs.3, Rs.2 and Rs.5, respectively. The assembly times per train at the three operations are 1, 3, and 1 minutes respectively. The corresponding times per train and per car are (2, 0, 4) and (1, 2, 0) minutes (a zero time indicates that the operation is not used). Determine the number of each toy assembled that maximizes the revenue.

Correlation and Regression

a) Calculate two regression coefficients when r=0.8, σX = 5and σy = 7. b) Find the probable error when number of items are 5 and correlation is 0.7 for a given distribution c) A student calculated the regression coefficient of x on y as - 1.25 and regression coefficient of y on x as - 2.40. Comment on his calculation. d) If n=8 and ∑d^2 =4 find rs? e) IF bxy= 0.8 and byx = 0.6 find ‘r’? f) Calculate two regression coefficients when r=0.8, σX = 5and σy = 7.

2. Calculate karl pearsons coefficient of correlation for the following data regarding price and demand of a certain commodity. Price: 21 22 23 24 25 26 27 28 29 Demand: 20 19 19 17 17 16 16 15 14
3. Compute karl pearson’s coefficient of correlation between per capita income and per capita consumer expenditure from the following data given below: Year: 1990 91 92 93 94 95 96 97 98 99 Per capita income: 249 251 248 252 258 269 271 272 280 275 Per capita Expenditure: 237 238 236 240 245 255 254 252 258 251
4. From the following table find correlation coefficient between age and playing habits of students: Age 15 16 17 18 19 20 No. of students 250 200 150 120 100 80 Regular players 200 150 90 48 30 12
5. The corresponding values of 2 series are given below in the table: ii) X: 42 44 58 55 89 98 66 iii) Y: 56 49 53 58 65 79 58 iv) Find the coefficient of correlation of the above series. How will you test the significance of ‘r’?
6. Calculate the karl pearson’s coefficient fro the following series of marks scored by ten students in a class test in Mathematics and Statistics. Marks in Mathematics 45 70 65 30 90 40 50 75 85 60 Marks in statistics 35 90 70 40 95 40 60 80 80 50
7. From the following calculate coefficient of correlation and probable error between advertisement expenses and sales of a company during the year 2008: i. Month jan feb mar apr may jun july aug sept oct nov dec ii. Adv. 50 60 70 90 120 150 140 160 170 190 200 150 iii. Sales 1200 1500 1600 2000 2200 2500 2400 2600 2800 2900 3100 3900
8. Calculate rank correlation coefficient between marks in statistics and accountancy. i. Marks in statistics 48 60 72 62 56 40 39 52 30 ii. Marks in accountancy 62 78 65 70 38 54 60 32 31
9. From the following data calculate coefficient of rank correlation between X and Y. X: 36 56 20 65 42 33 44 50 15 60 Y: 50 35 70 25 58 75 60 45 80 38
10. From the data given below compute Karl Pearsons coefficient of correlation: X series Y series Number of items 15 15

Mathematics in Finance

1 A machine was depreciated at the rate of 10% per annum for the first 2 years after it was purchased and at the rate of 13% per annum for the next four years and finally at the rate of 15% per annum for the rest. The original cost of the asset was Rs. 41,50,000. Find the scrap value of the asset at the end of seven years

2. The population of Delhi was 15,000 three years back, it is 22,000 right now; what will be the population three years later, if the rate of growth of population has been constant over the years and has been compounded annually?
3. Find the compound amount of Rs. 50000 for 4 years at 6%converted (a) annually (b) Semiannually (c) quarterly (d) Continuously 4 Find the time required for Rs 8000 to yield Rs 1250 at a simple interest of 5% per annum. 5 Find the rate of interest at which Rs 12000 will yield Rs 2060 in 2 years and 6 months.^ 6. 6 A person borrows Rs. 50,000 for six years at compound interest. The rate of interest is 7.5% per^ annum. What are the interest charges for years 3 and 5? What is the accumulated amount at the end of six year? 7 Balu borrowed Rs 25000 from Rahul but could not repay the amount in a period of 5 years. Accordingly, Rahul demands now Rs 35880 from Balu. At what percent p. a. compound interest did Rahul lend his money.

8 What amount lent at 10% p a compound interest will fetch Rs 630 as interest in 2 years. 3000 9 A sum of money invested at compound interest amounts to Rs 21632 in 2 years and to Rs 22497.28 in 3 years. Find the rate of interest and the sum invested.

10 The difference between the compound interest and the simple interest for 3 years at 5% p a on a certain sum of money was Rs 610. Find the sum.

11 An industry starts by producing 50000 units in its first year and the production for every year increases by 8% of that of the previous year. How many units will it produce in the seventh year? What is the sum total of the whole production in the first three years.? 12 A person has two daughters A and B aged 13 and 16 years. He has Rs 40000 with him now but wants that both of them should get an equal amount when they are 20 years old. How he should divide the money if it were to be deposited in a bank giving 9% compound interest per annum?

13 A certain sum of money at simple interest amounts to ₹560 in 3 years and to ₹600 in 5 years. Find the principal and the rate of interest. 14 At what rate percent will Rs 380 amount to Rs 893 in 15 years. 15 A banker^ borrows a certain sum at 8% p.a. compound interest compounded half yearly. He lends this money at 8% p.a. compound interest compounded quarterly. If he earns Rs. 9 in two years. Find the sum borrowed. 16 A loan of Rs. 2,25,000 to be repaid in two equal annual installments. If the rate of compound interest is 13% per annum, find the installment amount 17 The book value of the asset at the end of the 4th year is Rs. 10,00,000 and at the end of the 10th year is Rs. 5,31,441. If depreciation is written off under the declining balance method, find the rate of depreciation per annum and the original cost of the asset. 18 A manufacturer estimates that the value of his machinery depreciates by 13% of its value at the beginning of the year. Find the original value of the machine, if it depreciates by Rs. 5,655 during the second year 19 A machine was depreciated at the rate of 15% per annum for the first 2 years after it was purchased and at the rate of 10% for the next three years. The original cost of the asset was Rs. 20,000. Find the depreciated value of the asset at the end of five years. 20 A banker borrows a certain sum at 8% p.a. compound interest compounded half yearly. He lends this money at 8% p.a. compound interest compounded quarterly. If he earns Rs. 9 in two years. Find the sum

borrowed. 21 A loan of Rs. 2,25,000 to be repaid in^ two equal annual installments. If the rate of compound interest is 13% per annum, find the installment amount 22 The book value of the asset at the end of the 4th year is Rs. 10,00,000 and at the end of the 10th year is Rs. 5,31,441. If depreciation is written off under the declining balance method, find the rate of depreciation per annum and the original cost of the asset. 23 A manufacturer estimates that the value of his machinery depreciates by 13% of its value at the beginning of the year. Find the original value of the machine, if it depreciates by Rs. 5,655 during the second year Ordinary Annuity: When the payments are made at the end of the payment intervals, the annuity is called an ordinary annuity or annuity immediate. Annuity Due: When the payments are made at the beginning of the payment intervals, the annuity is called an annuity due 24 Find the present value and amount of an annuity due of Rs. 1500 payable once in the beginning^ of two months for 4 years, if the money is worth 12% compounded once in two months 25 Find the present worth of an annuity due of Rs. 500 payable at the beginning of each month for 2 years, if the money is worth 8% p a compounded monthly. 26 A bank pays 8% per year interest, compounded quarterly. What equal deposits have to be made at the beginning of each quarter for 10 years, to have Rs. 30,200 at the end of 10 years? 27 What amount should be set-aside at the beginning of each year to amount to Rs.38,688.12 at the end of 10 years at 14% per annum compounded annually? 28 Find the amount of an ordinary annuity of Rs.^ 3000 for 12 years at the rate of interest of 5% per annum. 29 Find the amount of an ordinary annuity of 15 monthly payments of Rs. 2500 that earn interest at 12% per annum compounded monthly 30 A man deposits Rs. 30000 at the end of each year for 20 years. He made his first payment at the end of year 1991 and last payment at the end of year 2011. How mush should be there in his account on 31 Dec. 2011, if the 10% interest rate is compounded annually. 31 A person is repaying a debt with payments of Rs.2,500 per^ month. If he misses his payments for July, August, September and October, what payment will be required in November to put him back on schedule, if interest is at 12.6% per annum? 32 A person is repaying a debt with payments of Rs.2,500 per month, at the beginning of every month. If he misses his payments for July, August, September and October, what payment will be required in November to put him back on schedule, if interest is at 12.6% per annum