A Poisson distribution with a mean of 1.84 results in about 45% of children being first born. In fact, the best you can do with any distribution is 54%. For example, if there are 10k couples, then there will be 18.4k children born, at most 10k of which can be first born; and 10/18.4 = 54%.

In fact, the only thing that matters for the distribution (once you know the mean) is the number of childless families, since they could have had a first born but didn't.

I haven't been able to figure out where 74% comes from. Is there some way to sample on a per-family basis to come up with this?

edit: spelling