Rolling forever

Probability Level 3

Suppose I have a line of squares, labelled $0,1,2,3,4, \ldots$ . I place a marker on the square 0. On every turn, I roll a die (with the numbers 1 to 6 and move forward as follows: if I was on square $P$ before the roll and I get a $x$ then I move to square $P+x$ . Let $X_n$ be the chance that I land on the square labelled $n$ . What is

$\displaystyle \lim_{n \rightarrow \infty} X_n ?$

\frac{1}{2}

\frac{1}{6}

\frac{1}{3}

\frac{2}{7}

1 solution

Lukas Riegler
Jan 13, 2019

Intuitive solution: The average number of squares the marker moves with each roll is $\frac{7}{2}$ .

So (in the long run) the marker will visit around $\frac{2}{7}$ of the squares. As $n\to\infty$ the probability converges to $\frac{2}{7}$ .

Rolling forever

1 solution

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