Probability that top k elements will be got?

Suppose $n$ vaiables $a_1$ ~ $a_n$ distributed over $[1,n]$. Without loss of generity, let $a_i =n- i+1$.

Suppose that each variable has a probability $p_e$ to be wrong, and the wrong value uniformly distributes over $[a_i-∝，a_i+∝]$ where $∝$ is far smaller than $n$.

If we take out $b$ largest variableswhere $b≥k$ and $b<<n$, the question is what the probability that $a_1$~$a_k$ are all in the set of $b$ variables?

• Variations of this have been asked before with Gaussian noise, no delta mass at the original values, and original values not in an arithmetic progression. I don't know that there is any simple answer, but how attached are you to this particular model? If there is no complete formula for everything, then you might need to focus on some part of the range of values. Which region is of particular interest? – Douglas Zare Jan 13 '15 at 4:26