# Why is Poisson instead of Negative Binomial used for computing attacker's potential progress?

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?

I think you are correct, since the success/failure of each nonce to satisfy the difficulty target is binary, rather than continuous, this is a discrete probability distribution and the correct distribution should be the negative binomial. However since the number of trials required per block is extremely large, ($> 10^20$ as of this writing), it is a very good approximation to take the limit $r\to\infty$ as described on the Wikipedia page. This is equivalent to taking the probability of success in any one trial to be zero, while keeping the mean of the distribution fixed, and results in the Poisson distribution. So, the difference is roughly less than 10^-20 and far to small to be measurable.