MTH 105 A & C, Spring 2024 Markov Processes: Problems

1. A cell phone provider classifies its customers as Low users (less than 400 minutes per month) or High users (400 or more minutes per month). Studies have shown that 80% of the people who were Low users one month will be Low users the next month, and that 70% of the people who were High users one month will be High users the next month.

(a) Suppose that during the month of January, 50% of customers were Low users and 50% were High users. What percent of customers will be Low users in February, and what percent will be High users?

(b) What about in March?

(c) Find the percents of customers that must be Low and High users respectively in order for these percents to remain steady from month to month.

2. For a certain group of states, it has been observed that 70% of Democratic governors are succeeded by Democrats, and 30% by Republicans. Also, 40% of Republican governors are succeeded by Democrats, and 60% by Republicans.

(a) Suppose that all current governors in these states are Democrats. (So, 100% Democrats and 0% Republicans.) Assuming that the pattern of transitions, de- scribed above, holds for the next two elections, what percent of governors will be Democrats after the second election?

(b) Find the percents of governors in the group of states that must be Democrats and Republicans respectively in order for the percents to remain steady from election cycle to election cycle.