Estimating the sum of subset of positive integers

Suppose we have a sequence of $n$ positive integers, each with an unknown value $c_1,c_2,c_3,...,c_n$ $c_i\in\mathbb N$

Note that each $c_i$ is sampled from a pmf: $p(c)$

The sum of these integers is a known constant $k$

$\sum_{i=1}^n c_i = k$

Now, the first m numbers are summed to give $x$

$\sum_{i=1}^m c_i = x$

How can we estimate $x$ (with $X$ ), such that the estimate is both as close to the actual value as possible, but also below the actual with probability at least $p$

$Pr(X<x) \geq p$

Apologies for shoddy notation, any help is greatly appreciated!

#NumberTheory

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$