# Calculating the expected number of items shared by chance in a Venn Diagram

I have a Venn Diagram that looks like this:

$$A) \, 213 \quad B) \, 160 \quad A\cap B) \, 100$$

The items from $A$ come from a population where their probability to be selected is $\frac{313}{12800}$.

The items from $B$ come from a population where their probability to be selected is $\frac{260}{1407}$.

How can I calculate a null model that tells me the number of shared items expected by chance if I create a Venn diagram for that data?

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Presumably the $213$ items are $A$ and not $B$ and the $160$ are $B$ and not $A$. However normally you would have one population, some of which has characteristic $A$ and some of which has characteristic $B$. Then the population outside both $A$ and $B$ would be the rest of the universe, here everything except $473$. Is that $12327$ or $934$? – Ross Millikan Feb 20 '13 at 18:36
The items (313 and 260) are expressed genes in a given condition. And the totals (12800 and 1407) are total sampled genes. What I want to prove is if this A & B is not a product of chance. – biojl Feb 21 '13 at 8:18

If $A$ and $B$ were independent, you would expect the same fraction of $B$'s to have $A$ as the whole population. Of the $260 B$'s, $100$ are also $A$'s, about $38\%$. But of the whole population, $A$'s are rare-a little less than $4\%$. Similarly the other way around. They are far from independent.