If possible, please write step by step. Thank you!! A sample of 2000 consumers i
ID: 3012825 • Letter: I
Question
If possible, please write step by step. Thank you!! A sample of 2000 consumers is grouped as follows: 600 users of Band A, 700 users of Brand B, and 700 who use neither brand. During any given month a Brand A user has a 10% probability of switching to Brand B and 30% probability of not using either brand. A Brand B user has a 20% probability of switching to Brand A and 10% probability of not using either brand. A nonuser has a 70% probability of purchasing Brand A and a 20% probability of purchasing Brand B. How many people from this sample will be in each category in one month? In two months? If x, y, z denote respectively the number of consumers which will be eventually using Brands A, B, and C in the long run, set up a linear system in x, y, z in the standard form whose solution would produce the steady state matrix bar X = [x y z]^T. Do not attempt to solve this system to find bar X, just set it up.Explanation / Answer
SOLUTION for both questions of Part (a)
[Note:
1. Probability figures are represented by decimal numbers, like 10% = 0.1
2. Remainder of switch over probability to other two brands from unity is taken as switch over probability for the brand.
3. Switch over figure = initial figure x probability of switch over]
FOR MONTH 1
Brand
Initial
figure
Switchover during the month
Brand A
Brand B
Neither
Brand A
600
600 x 0.6 = 360
600 x 0.1 = 60
600 x 0.3 = 180
Brand B
700
700 x 0.2 = 140
700 x 0.7 = 490
700 x 0.1 = 70
Neither
700
700 x 0.7 = 490
700 x 0.2 =140
700 x 0.1 = 70
Total
2000
990
690
320
Total figure above in bold gives the number of people in each brand in Month 1
FOR MONTH 2 [Final figure of Month 1 becomes the initial figure of Month 2]
Brand
Initial
figure
Switchover during the month
Brand A
Brand B
Neither
Brand A
990
990 x 0.6 = 594
990x 0.1 = 99
990x 0.3 = 297
Brand B
690
690x 0.2 = 138
690x 0.7 = 483
690x 0.1 = 69
Neither
320
320x 0.7 = 224
320x 0.2 = 64
320x 0.1 = 32
Total
956
646
398
Total figure above in bold gives the number of people in each brand in Month 2
NOTE: Total of final figure in both months must add to 2000, the initial total figure.
Brand
Initial
figure
Switchover during the month
Brand A
Brand B
Neither
Brand A
600
600 x 0.6 = 360
600 x 0.1 = 60
600 x 0.3 = 180
Brand B
700
700 x 0.2 = 140
700 x 0.7 = 490
700 x 0.1 = 70
Neither
700
700 x 0.7 = 490
700 x 0.2 =140
700 x 0.1 = 70
Total
2000
990
690
320
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