The following transition matrix represented customers upgrading their cell phones. Suppose that each customer upgtades to a new cell phone every two years. Use various powers of the transition matrix to find the probability that a customer who currently owns Phone A will select Phone A for the upgrades given in parts (a) through (c). Then use your results to answer part (d).
\[
\begin{array}{l}
\text { New Phone } \\
\text { A 、 B } \\
\text { Current Phone } \quad A\left[\begin{array}{rr}
0.4 & 0.6 \\
0.25 & 0.75
\end{array}\right] \\
\end{array}
\]
(a) Find the probability that a customer who currently owns Phone A selects Phone A with the first upgrade.
0.4 (Type an integer or decimal rounded to four decimal places as needed.)
(b) Find the probability that a customer who currently owns Phone A selects Phone A who the second upgrade.
0.31 (Type an integer or decimal rounded to four decimal places as needed.)
(c) Find the probability that a customer who currently owns Phone A selects Phone A with the third upgrade.
(Type an integer or decimal rounded to four decimal places as needed.)
Answer
Final Answer: The probability that a customer who currently owns Phone A will select Phone A for the third upgrade is \(\boxed{0.2965}\).
Step 1 ：The problem provides a transition matrix representing customers upgrading their cell phones. Each customer upgrades to a new cell phone every two years. The transition matrix is given as \(\left[\begin{array}{rr} 0.4 & 0.6 \ 0.25 & 0.75 \end{array}\right]\), where the first row represents the current state (owning Phone A) and the first column represents the next state (selecting Phone A).
Step 2 ：For part (a), the probability that a customer who currently owns Phone A selects Phone A with the first upgrade is directly given by the first entry of the matrix, which is 0.4.
Step 3 ：For part (b), the probability that a customer who currently owns Phone A selects Phone A with the second upgrade can be found by squaring the transition matrix and then looking at the entry in the first row and first column. The result is 0.31.
Step 4 ：For part (c), the probability that a customer who currently owns Phone A selects Phone A with the third upgrade can be found by cubing the transition matrix and then looking at the entry in the first row and first column. The result is 0.2965.
Step 5 ：Final Answer: The probability that a customer who currently owns Phone A will select Phone A for the third upgrade is \(\boxed{0.2965}\).
