What this kind probability should be called? I have $m$ continues integer points on a line, randomly uniform select $n$ points from the $m$ point without replacement. Order the points ascendingly. 
Let the random variable $A_i$ is the position (coordination on the line) of the $i$th point. So, $$P(A_i=k)=\frac{{k-1\choose i-1} {m-k \choose n-i}}{{m \choose n}} $$
How to derive the tail inequality for this probability.
The tail probability look something like this:
$$P(|A_i - E(A_i)| > t) < \sigma$$
I want the bound ($\sigma$) to be as tight as possible. The Chebyshev inequality is too loose. 
Updated:
Some supplement about the question: http://www.randomservices.org/random/urn/OrderStatistics.pdf 

Edit: See Didier's comment below. The binomial coefficients are "upside down" and so what's written below is meaningless. It is worthwhile, however, to see which tools are used to obtain tail estimates on the hypergeometric distribution, to get some ideas. Perhaps all they do is use Stirling's approximation and integrate it.
Your distribution is very close to a hypergeomtric distribution (as noted in an earlier version of the question). In fact, it is related to it via a factor of $i/k$. So tail estimates for it should transfer to tail estimates for your distribution. 
