IB Maths AI HL 4.19 Transition matrices and Markov chains Question Bank

Practise IB Mathematics HL 4.19 by applying transition matrices and markov chains methods to exam-style questions.

[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

\binom{X_{n+1}}{Z_{n+1}}=\boldsymbol{M}\binom{X_n}{Z_n}

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

(a)

What does b represent in this context?

[ 1 ]

(b)

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

[ 4 ]

(i)

when n=5;

[ 2 ]

$\left(\begin{smallmatrix}0.94&0.02\\0.06&0.98\end{smallmatrix}\right)^5\binom10$ OR $\left(\begin{smallmatrix}0.744&0.0852\\0.256&0.915\end{smallmatrix}\right)\binom10$ $0.744\ (0.744311\ldots)$
Note: Condone omission of the initial state vector for the M1.

(ii)

in the long term.

[ 2 ]

$\binom{0.25}{0.75}$
Note: Award A1 for $\binom{0.25}{0.75}$ OR $\left(\begin{smallmatrix}0.25&0.25\\0.75&0.75\end{smallmatrix}\right)$ seen.

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AHL 4.19 (HL)—Transition matrices and Markov chains question 1

Question (a)

Question (b)

Question (i)

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