IB Maths AI HL Ahl 3 15 Hl Adjacency Matrices and Transition Matrices Topic Practice

Question 1

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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 nNn\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).

Write down the value of b.

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