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IB Maths AI HL Ahl 4 19 Hl Transition Matrices and Markov Chains Questions

Practise IB Mathematics HL 4.19 by applying transition matrices and markov chains methods to questions from the Question Bank.

Syllabus
First assessment 2021
Course
Mathematics: applications and interpretation HL
Level
HL

Exam points

  • Construct transition diagrams, matrices and state vectors, then calculate s_n=T^n s_0.
  • Use powers of a transition matrix to analyse a discrete dynamical system or Markov chain.
  • Find and interpret steady states, long-term probabilities and the eigenvector/eigenvalue-1 relationship.

IB Maths AI HL Ahl 4 19 Hl Transition Matrices and Markov Chains Questions question 1

[Maximum number: 5]

A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.
The model is of the form

(Xn+1Zn+1)=M(XnZn)\binom{X_{n+1}}{Z_{n+1}}=\boldsymbol{M}\binom{X_n}{Z_n}

where XnX_n is the probability of the gene being in its normal state after dividing for the nth time, and ZnZ_n is the probability of it being in another state after dividing for the nth time, where n∈Nn\in\mathbb{N}.
Matrix M\boldsymbol{M} is found to be (0.94b0.060.98)\left(\begin{smallmatrix}0.94&b\\0.06&0.98\end{smallmatrix}\right).

Question (a)

(a)

What does b represent in this context?

[ 1 ]

Question (b)

(b)

The gene is in its normal state when n=0. Calculate the probability of it being in its normal state

[ 4 ]

Question (i)

(i)

when n=5;

[ 2 ]

Question (ii)

(ii)

in the long term.

[ 2 ]
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