An extensive survey of gym users in Metropolis indicates that 68% of the current
ID: 2901448 • Letter: A
Question
An extensive survey of gym users in Metropolis indicates that 68% of the current members of Silver's will continue their annual membership when it expires, 13% will quit and join Fitness Lab, and the rest will quit and join ThinNFit. Fitness Lab has been unable to keep its equipment in good shape, and as a result 32% of its members will defect to Silver's, and 34% will leave for ThinNFit. ThinNFit's members are quite happy, and as a result, 96% plan on renewing their annual membership, with half of the rest planning on moving to Silver's and half to Fitness Lab. Make a long-range prediction of the ultimate market shares of the three health clubs, assuming that current trends continue. (Round the answers to one decimal places.)
Silver ?%
Fitness Lab ?%
ThinNFit ?%
Explanation / Answer
Transition Matrix (T) :
Silver
Fitness Lab
ThinNFit
Silver
0.68
0.13
0.19
Fitness Lab
0.32
0.34
0.34
ThinNFit
0.02
0.02
0.96
Let the long term matrix be:
P = [p1 p2 p3]
We know,
P*T = P
On multiplying the matrix, we get
0.68*p1 + 0.32*p2 + 0.02*p3 = p1
0.13*p1 + 0.34*p2 + 0.02*p3 = p2
0.19*p1 + 0.34*p2 + 0.96*p3 = p3
No solution is possible in the long run.
Silver
Fitness Lab
ThinNFit
Silver
0.68
0.13
0.19
Fitness Lab
0.32
0.34
0.34
ThinNFit
0.02
0.02
0.96
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