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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 2

For hitting zero before four, the success probabilities are
$$
h_i=\frac{4-i}{4}.
$$
Conditioning each transition on eventual success gives
$$
\begin{array}{c|cc}
\text{state}&\text{down}&\text{up}\\ \hline
1&2/3&1/3\\
2&3/4&1/4\\
3&1&0.
\end{array}
$$
Thus the conditional expected hitting times satisfy
$$
e_1=1+\frac13e_2,
\qquad
e_2=1+\frac34e_1+\frac14e_3,
\qquad
e_3=1+e_2.
$$
Solving this linear system gives
$$
\boxed{\mathbb E(T\mid B)=e_1=\frac73}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 10E / b / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 10E / b / iii / Solution

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