Given an positive integer (whole number) $n$, let $s(n)$ be the sum of digits in its decimal expansion – that is, if $n=127$, then $s(n)=1+2+7=10$. Given another number $m$, we might ask how $s(n+m)$ differs from $s(n)$.

If $n = 127$ and $m = 25$ then $s(n+m)-s(n)= 1+5+2-(1+2+7)=-2$.

If $n = 132$ and $m = 25$ then $s(n+m)-s(n)= 1+5+7-(1+3+2)=7$.

Imelda randomly chooses $2023$–digit number $n$. What is the probability that $s(n+25)-s(n)=-11$? Give the answer as a decimal rounded to three decimal places.

Mathsbombe Competition 2023 is organised by the The Department of Mathematics at The University of Manchester.
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