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 survey 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) Write 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?​

Answer 1

The transition diagram is given in the attached image. See the definition of a transition diagram below.

What is a transition diagram?

A specific type of flowchart used for language analysis is called a transition diagram.

The boxes of the flowchart are represented as circles and are referred to as states in the transition diagram.

What is the Transition Matrix?

The transition matrix is given as follows:

$\left[\begin{array}{cc}0.6&0.4\\0.4&0.6\end{array}\right]$

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?​

If 10% of the people are using Brand X, then 90% are not using brand x. thus the initial state matrix is:

S₀ = [ 0.1, 0.9]. Thus one week later, the state matrix is:

S₁ = S₀P =  [0.1, 0.9]   *   $\left[\begin{array}{cc}0.6&0.4\\0.4&0.6\end{array}\right]$  

c11 = 0.1 x 0.6 + 0.9 x 0.6 = 0.6

c12 = 0.1 x 0.4 + 0.9 x 0.4 = 0.4

Hence

S₁  = [0.6, 0.4]

This means that the  percentage that will be using it 1 week later is:
0.38 = 38%

Learn more about Transition Diagrams at:
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