For those who think fingerprints are unique

Whenever I meet a new client, I always get asked the question, “Are fingerprints unique?” and before I can reply someone else replies, “Yes, they are absolute unique, I have heard that even the twins fingerprints are not similar and neither the fingerprints of two fingers of the same hand.”
While I agree with the answer, I am not sure about the accuracy of “Yes, they are absolutely unique” part. The reason is that we are about 6 billion of us. Assume that all 6 billion people have 10 fingers, that would make 60 billion fingerprints! Nobody has collected 60 billion fingerprints, matched against one another and proved the uniqueness! Does that mean that the fingerprints are not unique then? Well, no that would also be incorrect, because, fingerprints have been used for identification since 100 years now and we have yet to find 2 fingers with similar fingerprints. The correct answer to this question is:
The probability of 2 people having similar fingerprints is very low.