Overlapping sets (3) - significance test Sorry if this appears to be a naive question!
I have three overlapping sets and I want to find the probability of finding a larger/greater intersection for 'A intersect B intersect C' (in the example below, I want to find the probability of finding more than 135 elements that are common in sets A, B & C). For a two set problem, I guess I would do a Fisher or chi-square test. Here is what I have attempted so far:
### Prepare a 3 way contingency table:
mytable <- array(c(135,116,385,6256,
                   48,97,274,9555),
                 dim = c(2,2,2),
                 dimnames = list(
                    Is_C = c('Yes','No'),
                    Is_B = c('Yes','No'),
                    Is_A = c('Yes','No')))

## test
mantelhaen.test(myrabbit, exact = TRUE, alternative = "greater")

Is this the right test (alongwith the current parameters) to determine what I want or is there a more appropriate test for this?
 A: Let there be $n$ elements total with $a$, $b$, and $c$ in the subsets.  Consider $N$ already partitioned into $A$ and $N-A$.
The number of ways in which a $b$-element set can be drawn which has $l$ elements in common with $A$ is equal to the number of $l$-element subsets of $A$ times the number of $b-l$ element subsets of $N-A$:
$$\binom{a}{l}\binom{n-a}{b-l}.$$
Ensuant to that, the number of ways in which a $c$-element set can be drawn which has $m$ elements in common with $A\cap B$ is the number of $m$-element subsets of $A\cap B$ times the number of $c-m$-element subsets of $N-(A\cap B)$:
$$\binom{l}{m}\binom{n-l}{c-m}.$$
The total number of such choices of $B$ and $C$ is
$$\binom{n}{b}\binom{n}{c}$$
and they are all equally likely.  Summing over the possible values of $l$ gives the probability that the mutual intersection $A\cap B\cap C$ has exactly $m$ elements:
$$\Pr(m) = \frac{1}{\binom{n}{b} \binom{n}{c}}\sum _{l}\binom{a}{l} \binom{n-a}{b-l} \binom{l}{m} \binom{n-l}{c-m}.$$
(The sum extends over all values of $l$ that make sense; by defining $\binom{u}{v}=0$ whenever $v\lt 0$ or $v \gt u$, we do not need to indicate explicit endpoints.)

For instance, here is a plot of the central part of the probability distribution (comprising $99.99$% of the total) for $n=500, a=260, b=320, c=430$:

By summing these values from a particular value $m_0$ on up, we obtain the probability that the cardinality of the triple intersection equals or exceeds $m_0$.
In many practical applications, a Normal approximation to this sum works well.  As a demonstration, here are the log ratios of the true probabilities to those obtained with a Normal approximation.  (The Normal approximation uses the true mean $\mu$ and standard deviation $\sigma$ and estimates the probability of $m$ as $\Phi(\frac{m+1/2-\mu}{\sigma}) - \Phi(\frac{m-1/2-\mu}{\sigma})$ where $\Phi$ is the standard Normal CDF.)

The mean is near $143$ and the SD is near $5.9$.  (As usual, these are computed by summing $m$ and $m^2$, as weighted by the probabilities, to obtain the raw first and second moments, etc. I have not found a way to estimate either value a priori using simple formulas, but they can be estimated by computing probabilities for a small carefully-chosen selection of values of $m$ and fitting a Normal curve to them.)  Evidently, the approximation is excellent (in this case) within one SD of the mean, after which it starts decreasing, but even at three SD from the mean (i.e., around $125$ or $161$) it is still within one or two percent of the correct value.  For instance, the Normal approximation suggests the chance that $m \le 135$ is $0.939$ whereas the correct value is $0.928$.
In cases where there is a narrow range of possibilities of $m$, the distribution is far from Normal--but fortunately then the summations are short.  For instance, here is the distribution for $n=145, a=b=c=140$:

A simulation of $10^5$ independent instances of this situation tallies closely:

