A market survey shows that one third of the owners of Sorey State Boogie Boards became disenchanted with the product and switched to C&T Super Professional Boards the next surf season, while the other remained loyal to Sorey State. On the other hand, four fifths of the C&T Boogie Board users remained loyal to C&T, while the rest switched to Sorey State. Set these data up as a Markov transition matrix, and calculate the probability that a Sorey State Board user will be using the same brand two seasons later.
Take 1 = Sorey State, 2 = C&T.

A)The probability is $\frac { 3 } { 13 }$ .
B) The probability is $\frac { 11 } { 45 }$ .
C) The probability is $\frac { 23 } { 45 }$ .
D) The probability is $\frac { 29 } { 45 }$ .
E) The probability is $\frac { 16 } { 45 }$ .

Question

A market survey shows that one third of the owners of Sorey State Boogie Boards became disenchanted with the product and switched to C&T Super Professional Boards the next surf season, while the other remained loyal to Sorey State. On the other hand, four fifths of the C&T Boogie Board users remained loyal to C&T, while the rest switched to Sorey State. Set these data up as a Markov transition matrix, and calculate the probability that a Sorey State Board user will be using the same brand two seasons later.  
Take 1 = Sorey State, 2 = C&T.
 
A)The probability is $\frac { 3 } { 13 }$ .
B) The probability is $\frac { 11 } { 45 }$ .
C) The probability is $\frac { 23 } { 45 }$ .
D) The probability is $\frac { 29 } { 45 }$ .
E) The probability is $\frac { 16 } { 45 }$ .

Quizplus · Accepted Answer

The transition matrix is $\begin{bmatrix} \frac{2}{3} & \frac{1}{3} \ \frac{1}{5} & \frac{4}{5} \end{bmatrix}$. Calculating the square of this matrix gives the probability of staying with Sorey State after two seasons as $\frac{23}{45}$.

A Market Survey Shows That One Third of the Owners 313\frac { 3 } { 13 }133​

A Market Survey Shows That One Third of the Owners $\frac { 3 } { 13 }$