How do you determine if an occurrence in a subset is significant.

Question

There are roughly $22,000$ genes.
I have $1,200$ genes randomly chosen from the 22K in Group $A$.
I have $80$ genes in Group $B$ randomly chosen from the 22k.

How do I determine the probability of at least $2$ genes from group $B$ appearing in group $A$?

Answer 1

Suppose the $B-$genes are chosen. For a fixed $i$ the probability that the $A-$genes contain exactly $i$ $B-$genes is $$P_i=\frac {\binom {80}i\times \binom {22000-80}{1200-i}}{\binom {22000}{1200}}$$

The answer you seek is $$1-P_0-P_1\approx 0.9371388$$

So it is extremely likely that you'll have at least $2$ $B-$genes amongst the $A's$.

Worth noting that the probability that any given $A-$gene is also in $B$ is $\frac {80}{22000}$ so the expected number of $A-$genes in $B$ is $$\frac {80}{22000}\times 1200\approx 4.36$$