4.19 Transition matrices and Markov chainsIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 4.19 Transition matrices and Markov chains

1. Each day a commuter either cycles (C) or takes the bus (B). If she cycles today, the probability that she cycles tomorrow is $0.7$. If she takes the bus today, the probability that she cycles tomorrow is $0.4$. On Monday she cycles. Order the states C, B.
   Find the long-term probability that she cycles on a given day.2 marks
2. A machine is either working (W) or faulty (F) at the start of each day. A working machine is faulty the next day with probability $0.1$. A faulty machine is working the next day with probability $0.6$. The machine is working on day 0. Order the states W, F.
   Use your GDC to find the probability that the machine is faulty on day 3.2 marks
3. A website has $1000$ subscribers, each on one plan at the start of each month: Free (F), Standard (S) or Premium (P). Each month: a Free subscriber stays Free with probability $0.8$ and otherwise moves to Standard; a Standard subscriber moves to Free with probability $0.1$, stays with probability $0.7$ and otherwise moves to Premium; a Premium subscriber stays with probability $0.7$ and otherwise moves to Standard. Initially $600$ are Free, $300$ are Standard and $100$ are Premium. Order the states F, S, P.
   Write down the transition matrix $T$, and find the number of Premium subscribers after one month.3 marks