Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 4.19—transition Matrices – Markov Chains with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 4.19—transition Matrices – Markov Chains and mirrors Paper 1, 2, 3 style where relevant.
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A particle moves between three states, , , and , according to a Markov chain. The transition matrix is:
(a) Verify that is a valid transition matrix.
(b) Find the probability that the particle is in state after two transitions, given it starts in state .
A bicycle-sharing system has bicycles, each at either the East station or the West station at the end of a day. No bicycles enter or leave the system. The transition rule alternates between two types of day.
In the state order East, West, the column transition matrices are The first transition uses , the second uses , the third uses , and so on. Each bicycle follows the specified transition probabilities depending only on its current station and the type of day. Initially bicycles are at each station. Let be the column vector of expected counts after transitions, and let be its East component.
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A city transport office models the weekly travel mode of its commuters. Each week, every commuter travels by car (C), bus (B) or cycle (Y). The mode used next week depends only on the mode used this week.
Commuters behave independently, and the probabilities remain constant. State vectors are columns in the order C, B, Y, and . Unless otherwise stated, give answers exactly or correct to three significant figures.
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A population of mammals moves between regions and . In each annual migration, of the mammals in move to , and of those in move to . All movements in an annual migration are modelled simultaneously. There are no births, deaths or movements to other regions, and the transition probabilities remain constant.
Let be the model's expected population counts after annual migrations, with . Initially, and .
A calculator may be used.
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(e) Coffee costs $, tea $ and other drinks $. Find the long-run expected amount the customer spends per day, correct to the nearest cent.
(f) the validity of this model. Your answer should refer to two assumptions of a Markov chain in this context, and to the reliability of the estimated probabilities.
A café records the drink bought by one regular customer on each of consecutive days: coffee , tea or other . The table counts the day-to-day changes. Each row shows the drink on one day, and each column the drink on the following day.
The café models the customer's choices as a Markov chain with transition probabilities estimated from this table. State vectors are columns in the order and . Unless otherwise stated, give answers exactly or correct to three significant figures.
(b) The customer buys tea on a Monday. Find the probability that they buy tea on Wednesday. Find also the probability that they buy coffee on Tuesday and on Wednesday, and tea on Thursday.
(c) the eigenvalues of .
(a) the transition matrix .
(c) Find the exact steady- vector of .
(d) why the steady- vector equals the proportions of the recorded changes that start with each drink.
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