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

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Study AHL 4.19 (HL)—Transition matrices and Markov chains conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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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