In chapter 11 of the Satoshi's paper it is claimed that:
The recipient waits until the transaction has been added to a block and z blocks have been linked after it. He doesn't know the exact amount of progress the attacker has made, but assuming the honest blocks took the average expected time per block, the attacker's potential progress will be a Poisson distribution with expected value zq/p.
I think the precise model should be a negative binomial distribution instead of a Poisson one.
My question is: Is Satoshi's solution an approximation or is it supposed to describe accurately the problem?