A television campaign is conducted during the football season to promote a well-

Question

A television campaign is conducted during the football season to promote a well-known brand X shaving cream. For each of several weeks, a surven is made, and it is found that each week, 60% of those using brand X continue to use it, and 40% switch to another brand. It is also found that of those not using brand X, 40% switch to brand X while the other 60% continue using another brand. A) draw a transition diagram B) writhe the transition matrix C) if 10% of the people are using brand X at the start of the advertising campaign, what percentage will be using it 1 week later? 2 weeks later?

Explanation / Answer

A) Transition matrix P is:
Use X next week Not use X next week
Use X this week 0.60 0.40
Not Use X this week 0.40 0.60

B) 0.4*0.6+0.6*0.4=0.48; i.e., 48% will use brand X next week

C) Matrix P^2 is
0.52 0.48
0.48 0.52
So required probability is 0.4*0.52+0.6*0.48=0.496

D) Matrix P^3 is
0.408 0.496
0.496 0.408
So required probability is 0.4*0.408+0.6*0.496

E) Matrix P^n converges to the matrix
0.5 0.5
0.5 0.5

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