We could simulate an experiment with 10^5 random pairs:

In[2]:= N[Sum[a = RandomSample[Range[9]]; b = RandomSample[Range[9]];

Boole[Count[a - b, 0] == 2], {10^5}]/10^5]

Out[2]= 0.18364

That's not too far from the exact probability

In[3]:= (1/2) Sum[(-1)^j/j!, {j, 0, 7}]

Out[3]= 103/560

In[4]:= N[%]

Out[4]= 0.183929

See the following link for a proof:

**The Matching Problem**.