Cell phone users in Boise can choose between three carriers: Sprint, AT&T$,$ and Verizon. Advertising by all three companies results in a continual shift of customers. Each year, Sprint loses $30\%$ of customers to AT&T and $30\%$ to Verizon. AT&T loses $20\%$ to Sprint and $10\%$ to Verizon. Verizon loses $25\%$ to Sprint and $25\%$ to AT&T$.$ Currently, Sprint has $25\%$ of the market, AT&T has $35\%,$ and Verizon has $40\%.$

Use the order Sprint $($S$),$ AT&T $($A$),$ Verizon $($V$)$ throughout this assignment.

S$1.$ tate the probability vector: $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$2.$ Create the transition matrix, labeling rows and columns, and draw the transition diagram. Create your matrix with the rows ordered S$,$ A$,$ and V$,$ and columns ordered S$,$ A$,$ and V$.$

$3.$Assume that for $5$ years this trend continues. State the transition matrix after $5$ years.

$4.$What is the market share for each company after $5$ years?

Sprint: $\_\_\_\_\_\_\_\_\_\_\_\_\_\_$ AT&T: $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$ Verizon: $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$5.$ If the trend continues indefinitely, what will the market share be for each in the long run?