Is there any online calculator that can calculate
$$\dfrac{\sum_{k=570}^{770} \binom{6,700}{k}\binom{3,300}{1,000-k}}{\binom{10,000}{1,000}}
$$
for me? There are a few binomial coefficient calculators but for the sum in the numerator there are not usable.
Best Answer
Typing in the following input into WolframAlpha:
yields an exact result followed by the approximate decimal value, which to 50 digits is
$$0.99999999999855767906391784086205133574169750988724.$$
The same input in Mathematica will yield the same result as above, although you could equivalently use the input
As Jack D'Aurizio also suggested, this is a hypergeometric probability, so approximation by the normal distribution is possible. The way to do this is to recall the mean of a hypergeometric distribution with PDF $$\Pr[X = k] = \frac{\binom{m}{k}\binom{N-m}{n-k}}{\binom{N}{n}}$$ is given by $$\operatorname{E}[X] = \frac{mn}{N} = \mu,$$ and the variance is $$\operatorname{Var}[X] = \frac{mn(N-m)(N-n)}{N^2(N-1)} = \sigma^2.$$ In your case, you have values $m = 6700$, $N = 10000$, $n = 1000$. Thus we can approximate, with continuity correction, $$\begin{align*} \Pr[570 \le X \le 770] &\approx \Pr\left[\frac{570-\mu-0.5}{\sigma} \le \frac{X - \mu}{\sigma} \le \frac{770-\mu+0.5}{\sigma}\right]\\ &\approx \Pr[-7.12408 \le Z \le 7.12408]. \end{align*}$$ To fifty decimal places, this is $$0.99999999999895220978266454480005261506906978189913.$$ This approximation is good to about $10^{-13}$.